1,564
个编辑
更改
跳到导航
跳到搜索
Moved page from wikipedia:en:Pareto distribution (history)
此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
{{Probability distribution
{{Probability distribution
{概率分布
| name =Pareto Type I
| name =Pareto Type I
| name = Pareto Type i
| type =density
| type =density
类型 = 密度
| pdf_image =[[File:Probability density function of Pareto distribution.svg|325px|Pareto Type I probability density functions for various ''α'']]<br />Pareto Type I probability density functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math> As <math>\alpha \rightarrow \infty,</math> the distribution approaches <math>\delta(x - x_\mathrm{m}),</math> where <math>\delta</math> is the [[Dirac delta function]].
| pdf_image =Pareto Type I probability density functions for various α<br />Pareto Type I probability density functions for various \alpha with x_\mathrm{m} = 1. As \alpha \rightarrow \infty, the distribution approaches \delta(x - x_\mathrm{m}), where \delta is the Dirac delta function.
| pdf _ image = 各种 α < br/> Pareto i 型概率密度函数的 Pareto i 型概率密度函数。在 α 向右下方，分布趋近于 δ (x-x _ mathrm { m }) ，其中 δ 是狄拉克δ函数。
| cdf_image =[[File:Cumulative distribution function of Pareto distribution.svg|325px|Pareto Type I cumulative distribution functions for various ''α'']]<br />Pareto Type I cumulative distribution functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math>
| cdf_image =Pareto Type I cumulative distribution functions for various α<br />Pareto Type I cumulative distribution functions for various \alpha with x_\mathrm{m} = 1.
不同 α < br/> Pareto i 型累积分布函数的 Pareto i 型累积分布函数。
| parameters =<math>x_\mathrm{m} > 0</math> [[scale parameter|scale]] ([[real number|real]])<br /><math>\alpha > 0</math> [[shape parameter|shape]] (real)
| parameters =x_\mathrm{m} > 0 scale (real)<br />\alpha > 0 shape (real)
0 scale (real) < br/> alpha > 0 shape (real)
| support =<math>x \in [x_\mathrm{m}, \infty)</math>
| support =x \in [x_\mathrm{m}, \infty)
| support = x in [ x _ mathrm { m } ，infty)
| pdf =<math>\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}</math>
| pdf =\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}
1}{ x ^ { alpha + 1}}
| cdf =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha</math>
| cdf =1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha
1-left (frac { x _ mathrm { m }{ x } right) ^ alpha
| mean =<math>\begin{cases}
| mean =<math>\begin{cases}
开始{ cases }
\infty & \text{for }\alpha\le 1 \\
\infty & \text{for }\alpha\le 1 \\
1.1
\dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
\dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
1 & text { for } alpha > 1
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
| median =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math>
| median =x_\mathrm{m} \sqrt[\alpha]{2}
中位数 = x _ mathrm { m } sqrt [ alpha ]{2}
| mode =<math>x_\mathrm{m}</math>
| mode =x_\mathrm{m}
2009年10月11日
| variance =<math>\begin{cases}
| variance =<math>\begin{cases}
| 方差 = < math > begin { cases }
\infty & \text{for }\alpha\le 2 \\
\infty & \text{for }\alpha\le 2 \\
2.1.1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2
\dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2
\dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2
1) ^ 2(alpha-2)} & text { for } alpha > 2
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
| skewness =<math>\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3</math>
| skewness =\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3
| skewness = frac {2(1 + alpha)}{ alpha-3} sqrt { frac { alpha-2}{ alpha }}} text { for } alpha > 3
| kurtosis =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4</math>
| kurtosis =\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4
| 峭度 = frac {6(alpha ^ 3 + alpha ^ 2-6 alpha-2)}{ alpha (alpha-3)(alpha-4)} text { for } alpha > 4
| entropy =<math>\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right) </math>
| entropy =\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right)
| 熵 = log left (left (left (frac { x _ mathrm { m }{ alpha }右) ，e ^ {1 + tfrac {1}{ alpha }右)
| mgf =<math>\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0</math>
| mgf =\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0
| mgf = alpha (- x _ mathrm { m } t) ^ alpha Gamma (- alpha,-x _ mathrm { m } t) text { for } t < 0
| char =<math>\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)</math>
| char =\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)
| char = alpha (- ix _ mathrm { m } t) ^ alpha Gamma (- alpha,-ix _ mathrm { m } t)
| fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
| fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
| fisher = < math > mathcal { i }(x _ mathrm { m } ，alpha) = begin { bmatrix }
\dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\
\dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\
2} &-dfrac {1}{ x mathrm { m }
-\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2}
-\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2}
- dfrac {1}{ x _ mathrm { m } & dfrac {1}{ alpha ^ 2}
\end{bmatrix}</math>
\end{bmatrix}</math>
结束{ bmatrix } </math >
Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
右: < math > mathcal { i }(x _ mathrm { m } ，alpha) = begin { bmatrix }
\dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
\dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
2} & 0
0 & \dfrac{1}{\alpha^2}
0 & \dfrac{1}{\alpha^2}
0 & dfrac {1}{ alpha ^ 2}
\end{bmatrix}</math>
\end{bmatrix}</math>
结束{ bmatrix } </math >
}}
}}
}}
The '''Pareto distribution''', named after the Italian [[civil engineer]], [[economist]], and sociologist [[Vilfredo Pareto]],<ref>{{Cite journal|last=Amoroso|first=Luigi|date=1938|title=VILFREDO PARETO|url=|journal=Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest|volume=6|pages=|via=}}</ref> is a [[power-law]] [[probability distribution]] that is used in description of [[social sciences|social]], [[scientific]], [[geophysical]], [[actuarial science|actuarial]], and many other types of observable phenomena. Originally applied to describing the [[distribution of wealth]] in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population,<ref>{{Cite journal|last=Pareto|first=Vilfredo|date=1898|title=Cours d'economie politique|url=|journal=Journal of Political Economy|volume=6|pages=|via=}}</ref> the Pareto distribution has colloquially become known and referred to as the [[Pareto principle]], or "80-20 rule", and is sometimes called the "[[Matthew principle]]".<!-- can we find a better reference than https://youtu.be/5WX9UEYZsR8 at 2'10". --> This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, <math>\alpha</math> (''α'' = log<sub>4</sub>5 ≈ 1.16). While <math>\alpha</math> is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena<ref>{{Cite journal|last=VAN MONTFORT|first=M.A.J.|date=1986|title=The Generalized Pareto distribution applied to rainfall depths|journal=Hydrological Sciences Journal|volume=31|issue=2|pages=151–162|doi=10.1080/02626668609491037|doi-access=free}}</ref> and human activities.<ref>{{Cite journal|last=Oancea|first=Bogdan|date=2017|title=Income inequality in Romania: The exponential-Pareto distribution|journal=Physica A: Statistical Mechanics and Its Applications|volume=469|pages=486–498|doi=10.1016/j.physa.2016.11.094|bibcode=2017PhyA..469..486O}}</ref>
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. Originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population, the Pareto distribution has colloquially become known and referred to as the Pareto principle, or "80-20 rule", and is sometimes called the "Matthew principle".<!-- can we find a better reference than https://youtu.be/5WX9UEYZsR8 at 2'10". --> This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, \alpha (α = log45 ≈ 1.16). While \alpha is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena and human activities.
以意大利土木工程师、经济学家和社会学家 Vilfredo Pareto 命名的帕累托分布概率分布，是一种用于描述社会、科学、地球物理、保险精算和许多其他类型的可观测现象的幂定律。最初用于描述一个社会中的财富分配，符合大部分财富由一小部分人口持有的趋势，帕累托分布被通俗地称为帕雷托法则法则，或“80-20法则” ，有时也被称为“马太原则”。<！——我们能找到比2英尺10英寸的 https://youtu.be/5wx9ueyzsr8更好的参考吗。例如，这条规则规定，一个社会80% 的财富掌握在20% 的人口手中。然而，我们不应该把帕累托分布与帕雷托法则混为一谈，因为前者只对一个特定的幂值产生这个结果，即 α = log45≈1.16。虽然 α 值是一个参数，但经验观察发现80-20分布适用于各种情况，包括自然现象和人类活动。
==Definitions==
If ''X'' is a [[random variable]] with a Pareto (Type I) distribution,<ref name=arnold>{{cite book |author=Barry C. Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn= 978-0-89974-012-6|ref=harv}}</ref> then the probability that ''X'' is greater than some number ''x'', i.e. the [[survival function]] (also called tail function), is given by
If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by
如果 x 是一个带有 Pareto (Type i)分布的随机变量，那么 x 大于某个数 x 的概率，即。生存函数(也称为尾部函数)由
:<math>\overline{F}(x) = \Pr(X>x) = \begin{cases}
<math>\overline{F}(x) = \Pr(X>x) = \begin{cases}
< math > overline { f }(x) = Pr (x > x) = begin { cases }
\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
左(frac { x _ mathrm { m }{ x }右) ^ alpha & x ge x _ mathrm { m } ,
1 & x < x_\mathrm{m},
1 & x < x_\mathrm{m},
1 & x < x mathrm { m } ,
\end{cases}
\end{cases}
结束{ cases }
</math>
</math>
数学
where ''x''<sub>m</sub> is the (necessarily positive) minimum possible value of ''X'', and ''α'' is a positive parameter. The Pareto Type I distribution is characterized by a [[scale parameter]] ''x''<sub>m</sub> and a [[shape parameter]] ''α'', which is known as the ''tail index''. When this distribution is used to model the distribution of wealth, then the parameter ''α'' is called the [[Pareto index]].
where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
其中 xm 是 x 的(必然是正的)最小可能值，α 是一个正参数。帕累托 i 型分布的拥有属性是尺度参数 xm 和形状参数 α，即众所周知的尾部指数。当这个分布被用来模拟财富的分布时，参数 α 被称为帕累托指数。
===Cumulative distribution function===
From the definition, the [[cumulative distribution function]] of a Pareto random variable with parameters ''α'' and ''x''<sub>m</sub> is
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is
根据定义，带有参数 α 和 xm 的 Pareto 随机变量的累积分布函数是
:<math>F_X(x) = \begin{cases}
<math>F_X(x) = \begin{cases}
< math > f _ x (x) = begin { cases }
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
1-left (frac { x _ mathrm { m }{ x } right) ^ alpha & x ge x _ mathrm { m } ,
0 & x < x_\mathrm{m}.
0 & x < x_\mathrm{m}.
0 & x < x mathrm { m }.
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
===Probability density function===
It follows (by [[Derivative|differentiation]]) that the [[probability density function]] is
It follows (by differentiation) that the probability density function is
由此可以得出结论: 概率密度函数
:<math>f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases} </math>
f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases}
F _ x (x) = begin { cases } frac { alpha x mathrm { m } ^ alpha }{ x ^ alpha + 1} & x ge x mathrm { m } ，0 & x < x mathrm { m }.结束{ cases }
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes [[asymptotically]]. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a [[log-log plot]], the distribution is represented by a straight line.
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a log-log plot, the distribution is represented by a straight line.
在直线轴上绘制时，分布曲线呈现常见的 j 形曲线，渐近地接近每个正交轴。曲线的所有部分都是自相似的(取决于适当的比例因子)。在对数对数图中绘制时，分布由一条直线表示。
==Properties==
===Moments and characteristic function===
* The [[expected value]] of a [[random variable]] following a Pareto distribution is
:
:: <math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
<math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
(x) = begin { cases } infty & alpha le 1,
\frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha>1.
\frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha>1.
1.1.1.1.1.2.1.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
* The [[variance]] of a [[random variable]] following a Pareto distribution is
:: <math>\operatorname{Var}(X)= \begin{cases}
<math>\operatorname{Var}(X)= \begin{cases}
{ Var }(x) = begin { cases }
\infty & \alpha\in(1,2], \\
\infty & \alpha\in(1,2], \\
(1,2)中的 infty & alpha
\left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha>2.
\left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha>2.
左(frac { x _ mathrm { m }{ alpha-1}右) ^ 2 frac { alpha-2} & alpha > 2。
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
: (If ''α'' ≤ 1, the variance does not exist.)
(If α ≤ 1, the variance does not exist.)
(如果 α ≤1，方差不存在.)
* The raw [[moment (mathematics)|moments]] are
:: <math>\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}</math>
\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}
N’ = begin { cases } infty & alpha le n，frac { alpha x mathrm { m } ^ n }{ alpha-n } & alpha > n。结束{ cases }
* The [[Moment-generating function|moment generating function]] is only defined for non-positive values ''t'' ≤ 0 as
::<math>M\left(t;\alpha,x_\mathrm{m}\right) = \operatorname{E} \left [e^{tX} \right ] = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t)</math>
M\left(t;\alpha,x_\mathrm{m}\right) = \operatorname{E} \left [e^{tX} \right ] = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t)
M left (t; alpha，x _ mathrm { m } right) = operatorname { e } left [ e ^ { tX } right ] = alpha (- x _ mathrm { m } t) ^ alpha Gamma (- alpha,-x _ mathrm { m } t)
::<math>M\left(0,\alpha,x_\mathrm{m}\right)=1.</math>
M\left(0,\alpha,x_\mathrm{m}\right)=1.
M 左(0，alpha，x _ mathrm { m }右) = 1。
* The [[Characteristic function (probability theory)|characteristic function]] is given by
:: <math>\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),</math>
\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),
Varphi (t; alpha，x _ mathrm { m }) = alpha (- ix _ mathrm { m } t) ^ alpha Gamma (- alpha,-ix _ mathrm { m } t) ,
: where Γ(''a'', ''x'') is the [[incomplete gamma function]].
where Γ(a, x) is the incomplete gamma function.
其中 γ (a，x)是不完全Γ函数。
The parameters may be solved using the [[method of moments]].<!-- :
The parameters may be solved using the method of moments.<!-- :
参数可以用矩量法求解。 < ! -- :
alpha = 1 + sqr(1 + mean ^ 2 / var)
alpha = 1 + sqr(1 + mean ^ 2 / var)
Alpha = 1 + sqr (1 + mean ^ 2/var)
beta = mean * Sqr(mean ^ 2 + var) / (Sqr(var) + Sqr(mean ^ 2 + var)) -->
beta = mean * Sqr(mean ^ 2 + var) / (Sqr(var) + Sqr(mean ^ 2 + var)) -->
Beta = mean * Sqr (mean ^ 2 + var)/(Sqr (var) + Sqr (mean ^ 2 + var)) -- >
<ref>S. Hussain, S.H. Bhatti (2018).
[https://www.researchgate.net/publication/322758024_Parameter_estimation_of_Pareto_distribution_Some_modified_moment_estimators Parameter estimation of Pareto distribution: Some modified moment estimators]. ''Maejo International Journal of Science and Technology'' 12(1):11-27</ref>
===Conditional distributions===
The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number x_1 exceeding x_\text{m}, is a Pareto distribution with the same Pareto index \alpha but with minimum x_1 instead of x_\text{m}.
一个帕雷托分布的随机变量的条件概率分布，假设它大于或等于一个特定的数字 x 1超过 x _ text { m } ，是一个具有相同帕雷托指数 α 但最小 x _ 1而不是 x _ text { m }的帕累托分布。
The [[conditional probability distribution]] of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number <math>x_1</math> exceeding <math>x_\text{m}</math>, is a Pareto distribution with the same Pareto index <math>\alpha</math> but with minimum <math>x_1</math> instead of <math>x_\text{m}</math>.
===A characterization theorem===
Suppose X_1, X_2, X_3, \dotsc are independent identically distributed random variables whose probability distribution is supported on the interval [x_\text{m},\infty) for some x_\text{m}>0. Suppose that for all n, the two random variables \min\{X_1,\dotsc,X_n\} and (X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\} are independent. Then the common distribution is a Pareto distribution.
假设 x _ 1，x _ 2，x _ 3，dotsc 是独立同分布的随机变量，对于某些 x _ text { m } > 0，其概率分布在区间[ x _ text { m } ，infty ]上是支持的。假设对于所有 n，两个随机变量 min { x1，dotsc，xn }和(x1 + dotsb + xn)/min { x1，dotsc，xn }是独立的。那么公共分配就是帕累托分布。
Suppose <math>X_1, X_2, X_3, \dotsc</math> are [[independent identically distributed]] [[random variable]]s whose probability distribution is supported on the interval <math>[x_\text{m},\infty)</math> for some <math>x_\text{m}>0</math>. Suppose that for all <math>n</math>, the two random variables <math>\min\{X_1,\dotsc,X_n\}</math> and <math>(X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\}</math> are independent. Then the common distribution is a Pareto distribution.{{Citation needed|date=February 2012}}
===Geometric mean===
The geometric mean (G) is
几何平均(g)是
The [[geometric mean]] (''G'') is<ref name=Johnson1994>Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.</ref>
G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).
G = x _ text { m } exp left (frac {1}{ alpha } right).
: <math> G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).</math>
===Harmonic mean===
The harmonic mean (H) is of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions. Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto distribution generalizes Pareto Type IV.
调和平均数(h)是帕累托分布，称为帕累托 i 型、 II 型、 III 型、 IV 型和 Feller-帕累托分布。帕累托类型 IV 包含帕累托类型 i-III 作为特殊情况。Feller-帕累托分布推广了 Pareto 第四类。
The [[harmonic mean]] (''H'') is<ref name="Johnson1994"/>
<!--- In this context using x_m for the lower bound for the scale parameter is not meaningful, usual notation is \sigma --->
< ! ——在这种情况下，使用 x _ m 作为刻度参数的下限是没有意义的，通常的表示法是 sigma —— >
: <math> H = x_\text{m} \left( 1 + \frac{ 1 }{ \alpha } \right).</math>
The Pareto distribution hierarchy is summarized in the next table comparing the survival functions (complementary CDF).
下表总结了帕累托分布的层次结构，比较了生存函数(补充的 CDF)。
===Graphical representation===
The characteristic curved '[[long tail]]' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a [[log-log graph]], which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for ''x'' ≥ ''x''<sub>m</sub>,
When μ = 0, the Pareto distribution Type II is also known as the Lomax distribution.
当 μ = 0时，帕累托分布 II 型也称为洛马克斯分布。
:<math>\log f_X(x)= \log \left(\alpha\frac{x_\mathrm{m}^\alpha}{x^{\alpha+1}}\right) = \log (\alpha x_\mathrm{m}^\alpha) - (\alpha+1) \log x.</math>
In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.
在本节中，符号 xm (前面用来表示 x 的最小值)被 σ 替换。
Since ''α'' is positive, the gradient −(''α'' + 1) is negative.
{|class="wikitable" border="1"
{ | class = “ wikitable” border = “1”
|+Pareto distributions
| + 帕累托分布
==Related distributions==
! !! \overline{F}(x)=1-F(x) !! Support !! Parameters
!!!1-F (x) ! ！支持！参数
===Generalized Pareto distributions===
|-
|-
{{See also|Generalized Pareto distribution}}
| Type I
| 第一类
|| \left[\frac x \sigma \right]^{-\alpha}
[ frac x sigma right ] ^ {-alpha }
There is a hierarchy <ref name=arnold/><ref name=jkb94>Johnson, Kotz, and Balakrishnan (1994), (20.4).</ref> of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.<ref name=arnold/><ref name=jkb94/><ref name=kk03>{{cite book |author1=Christian Kleiber |author2=Samuel Kotz |lastauthoramp=yes |year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences |publisher=[[John Wiley & Sons|Wiley]] |isbn=978-0-471-15064-0|ref=harv| url=https://books.google.com/books?id=7wLGjyB128IC}}</ref> Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto<ref name=jkb94/><ref name=feller>{{cite book|last=Feller |first= W.| year=1971| title=An Introduction to Probability Theory and its Applications| volume=II| edition=2nd | location= New York|publisher=Wiley|page=50}} "The densities (4.3) are sometimes called after the economist ''Pareto''. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ ''Ax''<sup>−''α''</sup> as ''x'' → ∞."</ref> distribution generalizes Pareto Type IV.
|| x \ge \sigma
我们会找到他的
<!--- In this context using x_m for the lower bound for the scale parameter is not meaningful, usual notation is \sigma --->
|| \sigma > 0, \alpha
| | sigma > 0，alpha
|-
|-
====Pareto types I–IV====
| Type II
| 第二类
The Pareto distribution hierarchy is summarized in the next table comparing the [[survival function]]s (complementary CDF).
|| \left[1 + \frac{x-\mu} \sigma \right]^{-\alpha}
| | 左[1 + frac { x-mu } sigma right ] ^ {-alpha }
|| x \ge \mu
我们会找到他的
When ''μ'' = 0, the Pareto distribution Type II is also known as the [[Lomax distribution]].<ref>{{cite journal | last1 = Lomax | first1 = K. S. | year = 1954 | title = Business failures. Another example of the analysis of failure data | url = | journal = Journal of the American Statistical Association | volume = 49 | issue = 268| pages = 847–52 | doi=10.1080/01621459.1954.10501239}}</ref>
|| \mu \in \mathbb R, \sigma > 0, \alpha
数学天才，西格玛 > 0，阿尔法
|-
|-
In this section, the symbol ''x''<sub>m</sub>, used before to indicate the minimum value of ''x'', is replaced by ''σ''.
| Lomax
| 洛马克斯
|| \left[1 + \frac x \sigma \right]^{-\alpha}
[1 + frac x sigma right ] ^ {-alpha }
{|class="wikitable" border="1"
|| x \ge 0
我们会找到他的，我们会找到他
|+Pareto distributions
|| \sigma > 0, \alpha
| | sigma > 0，alpha
! !! <math> \overline{F}(x)=1-F(x)</math> !! Support !! Parameters
|-
|-
|-
| Type III
| 第三类
| Type I
|| \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1}
| | 左[1 + 左(frac { x-mu } sigma 右) ^ {1/gamma }右] ^ {-1}
|| <math>\left[\frac x \sigma \right]^{-\alpha}</math>
|| x \ge \mu
我们会找到他的
|| <math>x \ge \sigma</math>
|| \mu \in \mathbb R, \sigma, \gamma > 0
| | mu in mathbb r，sigma，gamma > 0
|| <math>\sigma > 0, \alpha</math>
|-
|-
|-
| Type IV
| 第 IV 类
| Type II
|| \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha}
| | 左[1 + 左(frac { x-mu } sigma 右) ^ {1/gamma }右] ^ {-alpha }
|| <math>\left[1 + \frac{x-\mu} \sigma \right]^{-\alpha}</math>
|| x \ge \mu
我们会找到他的
|| <math>x \ge \mu</math>
|| \mu \in \mathbb R, \sigma, \gamma > 0, \alpha
| | mu in mathbb r，sigma，gamma > 0，alpha
|| <math>\mu \in \mathbb R, \sigma > 0, \alpha</math>
|-
|-
|-
|-
|-
| Lomax
|}
|}
|| <math>\left[1 + \frac x \sigma \right]^{-\alpha}</math>
|| <math>x \ge 0</math>
The shape parameter α is the tail index, μ is location, σ is scale, γ is an inequality parameter. Some special cases of Pareto Type (IV) are
形状参数 α 是尾部指标，μ 是位置，σ 是标度，γ 是不等式参数。帕累托型(IV)的一些特殊情况是
|| <math>\sigma > 0, \alpha</math>
|-
P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),
P (IV)(sigma，sigma，1，alpha) = p (i)(sigma，alpha) ,
| Type III
P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),
P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1} </math>
P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).
p (IV)(mu，sigma，gamma，1) = p (III)(mu，sigma，gamma).
|| <math>x \ge \mu</math>
|| <math> \mu \in \mathbb R, \sigma, \gamma > 0</math>
The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments are finite for some δ > 0, as shown in the table below, where δ is not necessarily an integer.
均值的有限性、方差的存在性和有限性取决于尾指数 α (不等式指数 γ)。特别是，对于某些 δ > 0，分数阶 δ- 矩是有限的，如下表所示，其中 δ 不一定是整数。
|-
| Type IV
{|class="wikitable" border="1"
{ | class = “ wikitable” border = “1”
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha}</math>
|+Moments of Pareto I–IV distributions (case μ = 0)
帕累托 i-IV 分布的 | + 矩(μ = 0)
|| <math>x \ge \mu</math>
! !! \operatorname{E}[X] !! Condition !! \operatorname{E}[X^\delta] !! Condition
!!!操作员名称{ e }[ x ] ! ！条件！操作员名称{ e }[ x ^ delta ] ！环境影响评估条件
|| <math>\mu \in \mathbb R, \sigma, \gamma > 0, \alpha</math>
|-
|-
|-
| Type I
| 第一类
|-
|| \frac{\sigma \alpha}{\alpha-1}
| | frac { sigma alpha }{ alpha-1}
|}
|| \alpha > 1
| | alpha > 1
|| \frac{\sigma^\delta \alpha}{\alpha-\delta}
| | frac { sigma ^ delta alpha }{ alpha-delta }
The shape parameter ''α'' is the [[tail index]], ''μ'' is location, ''σ'' is scale, ''γ'' is an inequality parameter. Some special cases of Pareto Type (IV) are
|| \delta < \alpha
| | delta < alpha
|-
|-
::<math> P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),</math>
| Type II
| 第二类
::<math> P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),</math>
|| \frac{ \sigma }{\alpha-1}
| | frac { sigma }{ alpha-1}
::<math> P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).</math>
|| \alpha > 1
| | alpha > 1
|| \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}
| | frac { sigma ^ delta Gamma (alpha-delta) Gamma (1 + delta)}{ Gamma (alpha)}
The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index ''α'' (inequality index ''γ''). In particular, fractional ''δ''-moments are finite for some ''δ'' > 0, as shown in the table below, where ''δ'' is not necessarily an integer.
|| 0 < \delta < \alpha
| | 0 < delta < alpha
|-
|-
{|class="wikitable" border="1"
| Type III
| 第三类
|+Moments of Pareto I–IV distributions (case ''μ'' = 0)
|| \sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)
| | sigma Gamma (1-Gamma) Gamma (1 + Gamma)
! !! <math>\operatorname{E}[X]</math> !! Condition !! <math>\operatorname{E}[X^\delta]</math> !! Condition
|| -1<\gamma<1
| |-1 < γ < 1
|-
|| \sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)
| | sigma ^ delta Gamma (1-Gamma delta) Gamma (1 + Gamma delta)
| Type I
|| -\gamma^{-1}<\delta<\gamma^{-1}
| |-gamma ^ {-1} < delta < gamma ^ {-1}
|| <math>\frac{\sigma \alpha}{\alpha-1}</math>
|-
|-
|| <math>\alpha > 1</math>
| Type IV
| 第 IV 类
|| <math>\frac{\sigma^\delta \alpha}{\alpha-\delta}</math>
|| \frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)}
| | frac { sigma Gamma (alpha-Gamma) Gamma (1 + Gamma)}{ Gamma (alpha)}
|| <math> \delta < \alpha</math>
|| -1<\gamma<\alpha
| |-1 < gamma < alpha
|-
|| \frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}
| | frac { sigma ^ delta Gamma (alpha-Gamma delta) Gamma (1 + Gamma delta)}{ Gamma (alpha)}
| Type II
|| -\gamma^{-1}<\delta<\alpha/\gamma
| |-gamma ^ {-1} < delta < alpha/gamma
|| <math> \frac{ \sigma }{\alpha-1}</math>
|-
|-
|| <math>\alpha > 1</math>
|-
|-
|| <math> \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}</math>
|}
|}
|| <math>0 < \delta < \alpha</math>
|-
| Type III
Feller
费勒
|| <math>\sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)</math>
|| <math> -1<\gamma<1</math>
W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma
W = mu + sigma left (frac { u _ 1}{ u _ 2} right) ^ gamma
|| <math>\sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)</math>
|| <math>-\gamma^{-1}<\delta<\gamma^{-1}</math>
and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are
写出 w ~ FP (μ，σ，γ，δ1，δ2)。帕累托分布的特殊情况如下
|-
| Type IV
FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)
FP (sigma，sigma，1,1，alpha) = p (i)(sigma，alpha)
|| <math>\frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)}</math>
FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)
FP (mu，sigma，1,1，alpha) = p (II)(mu，sigma，alpha)
|| <math> -1<\gamma<\alpha</math>
FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)
FP (mu，sigma，gamma，1,1) = p (III)(mu，sigma，gamma)
|| <math>\frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}</math>
FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).
FP (mu，sigma，gamma，1，alpha) = p (IV)(mu，sigma，gamma，alpha).
|| <math>-\gamma^{-1}<\delta<\alpha/\gamma </math>
|-
|-
The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum xm and index α, then
美国帕累托分布协会与美国指数分布协会有如下关系。如果 x 是以最小 xm 和指数 α 为参数的 pareto 分布，则
|}
Y = \log\left(\frac{X}{x_\mathrm{m}}\right)
Y = log left (frac { x }{ x _ mathrm { m } right)
====Feller–Pareto distribution====
Feller<ref name=jkb94/><ref name=feller/> defines a Pareto variable by transformation ''U'' = ''Y''<sup>−1</sup> − 1 of a [[beta distribution|beta random variable]] ''Y'', whose probability density function is
is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then
指数分布的速率参数 α。等价，如果 y 是指数分布的速率 α，则
:<math> f(y) = \frac{y^{\gamma_1-1} (1-y)^{\gamma_2-1}}{B(\gamma_1, \gamma_2)}, \qquad 0<y<1; \gamma_1,\gamma_2>0,</math>
x_\mathrm{m} e^Y
2. x mathrm { m } e ^ y
where ''B''( ) is the [[beta function]]. If
is Pareto-distributed with minimum xm and index α.
是以最小 xm 和指数 α 为参数的 pareto 分布。
:<math> W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma>0, \gamma>0,</math>
This can be shown using the standard change-of-variable techniques:
这可以使用标准的变量更改技术来显示:
then ''W'' has a Feller–Pareto distribution FP(''μ'', ''σ'', ''γ'', ''γ''<sub>1</sub>, ''γ''<sub>2</sub>).<ref name=arnold/>
<math>
《数学》
\begin{align}
开始{ align }
If <math>U_1 \sim \Gamma(\delta_1, 1)</math> and <math>U_2 \sim \Gamma(\delta_2, 1)</math> are independent [[Gamma distribution|Gamma variables]], another construction of a Feller–Pareto (FP) variable is<ref>{{cite book |last=Chotikapanich |first=Duangkamon |title=Modeling Income Distributions and Lorenz Curves |chapter=Chapter 7: Pareto and Generalized Pareto Distributions |date=16 September 2008 |pages=121–22 |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}</ref>
\Pr(Y<y) & = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) \\
Pr (y < y) & = Pr left (log left (frac { x }{ x _ mathrm { m } right) < y right)
& = \Pr(X<x_\mathrm{m} e^y) = 1-\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1-e^{-\alpha y}.
1-left (frac { x _ mathrm { m } e ^ y }{ x _ mathrm { m } e ^ y } right) ^ alpha = 1-e ^ {-alpha y }.
:<math>W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma</math>
\end{align}
结束{ align }
</math>
数学
and we write ''W'' ~ FP(''μ'', ''σ'', ''γ'', ''δ''<sub>1</sub>, ''δ''<sub>2</sub>). Special cases of the Feller–Pareto distribution are
The last expression is the cumulative distribution function of an exponential distribution with rate α.
最后一个表达式是速率为 α 的累积分布函数的指数分布。
:<math>FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)</math>
:<math>FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)</math>
:<math>FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)</math>
The Pareto distribution and log-normal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the exponential distribution and normal distribution. (See the previous section.)
帕累托分布和对数正态分布是描述相同类型数量的替代分布。这两者之间的联系之一是，它们都是指数分布的随机变量，分别按照其他常见的分布，即指数分布和正态分布。(见上一节)
:<math>FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).</math>
===Relation to the exponential distribution===
The Pareto distribution is a special case of the generalized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the Lomax distribution as a special case. This family also contains both the unshifted and shifted exponential distributions.
帕累托分布分布是广义帕累托分布分布的一个特例，它是一族形式相似的分布，但包含一个额外的参数，使得分布的支撑要么在下面有界(在一个可变点) ，要么在上面和下面都有界(两者都是可变的) ，洛马克斯分布是一个特例。该族同时包含无移位和移位指数分布。
The Pareto distribution is related to the [[exponential distribution]] as follows. If ''X'' is Pareto-distributed with minimum ''x''<sub>m</sub> and index ''α'', then
The Pareto distribution with scale x_m and shape \alpha is equivalent to the generalized Pareto distribution with location \mu=x_m, scale \sigma=x_m/\alpha and shape \xi=1/\alpha. Vice versa one can get the Pareto distribution from the GPD by x_m = \sigma/\xi and \alpha=1/\xi.
具有尺度 x _ m 和形状 α 的帕累托分布相当于位置 mu = x _ m，尺度 σ = x _ m/α，形状 xi = 1/α 的广义帕累托分布。反之亦然，人们可以通过 x _ m = sigma/xi 和 α = 1/xi 从 GPD 得到帕累托分布。
: <math> Y = \log\left(\frac{X}{x_\mathrm{m}}\right) </math>
is [[exponential distribution|exponentially distributed]] with rate parameter ''α''. Equivalently, if ''Y'' is exponentially distributed with rate ''α'', then
{{Probability distribution
{概率分布
: <math> x_\mathrm{m} e^Y</math>
| name =Bounded Pareto
| name = Bounded Pareto
| type =density
类型 = 密度
is Pareto-distributed with minimum ''x''<sub>m</sub> and index ''α''.
| pdf_image =
图片来源: pdf
| cdf_image =
图片 | cdf/image =
This can be shown using the standard change-of-variable techniques:
| parameters =
参数 =
L > 0 location (real)<br />
L > 0 location (real) < br/>
: <math>
H > L location (real)<br />
H > l 位置(real) < br/>
\begin{align}
\alpha > 0 shape (real)
α > 0形状(实数)
\Pr(Y<y) & = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) \\
| support =L \leqslant x \leqslant H
| support = l leqslanx leqslanh
& = \Pr(X<x_\mathrm{m} e^y) = 1-\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1-e^{-\alpha y}.
| pdf =\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}
1-left (frac { l }{ h } right) ^ alpha }
\end{align}
| cdf =\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}
| cdf = frac {1-L ^ alpha x ^ {-alpha }{1-left (frac { l }{ h } right) ^ alpha }
</math>
| mean =
意味着
\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1 <br />
左(frac { l ^ alpha }{1-left (frac { l }{ h } right) ^ alpha } cdot left (frac { alpha }{ alpha-1} right) cdot left (frac {1}{ l ^ { alpha-1}-frac {1}{ h ^ { alpha-1}} right) ，alpha neq 1 < br/>
The last expression is the cumulative distribution function of an exponential distribution with rate ''α''.
\frac\ln\frac{H}{L}, \alpha=1
1.1.1
| median = L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}
| median = l left (1-frac {1}{2} left (1-left (1-left (frac { l }{ h } right) ^ alpha right)) ^ {-frac {1}{ alpha }}
===Relation to the log-normal distribution===
| mode =
2012年10月22日
The Pareto distribution and [[log-normal distribution]] are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the [[exponential distribution]] and [[normal distribution]]. (See [[#Relation_to_the_exponential_distribution|the previous section]].)
| variance =
| 方差 =
\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2
左(frac { l ^ alpha }{1-left (frac { l }{ h } right) ^ alpha } cdot left (frac { alpha }{ alpha-2} right) cdot left (frac {1}{ l ^ { alpha-2}-frac {1}{ h ^ { alpha-2}} right) ，alpha neq 2
===Relation to the generalized Pareto distribution===
\frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2
2{ h } ^ 2{ l } ^ 2}{ h } ^ 2-{ l } ^ 2} ln frac { h }{ l } ，alpha = 2
The Pareto distribution is a special case of the [[generalized Pareto distribution]], which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the [[Lomax distribution]] as a special case. This family also contains both the unshifted and shifted [[exponential distribution]]s.
(this is the second raw moment, not the variance)
(这是第二个原始时刻，不是方差)
| skewness = \frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha * (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j
| skewness = frac { l ^ { alpha }{1-left (frac { l }{ h } right) ^ { alpha } cdot frac { alpha * (l ^ { k-alpha }-h ^ { k-alpha })}{(alpha-k)}} ，alpha neq j
The Pareto distribution with scale <math>x_m</math> and shape <math>\alpha</math> is equivalent to the generalized Pareto distribution with location <math>\mu=x_m</math>, scale <math>\sigma=x_m/\alpha</math> and shape <math>\xi=1/\alpha</math>. Vice versa one can get the Pareto distribution from the GPD by <math>x_m = \sigma/\xi</math> and <math>\alpha=1/\xi</math>.
(this is the kth raw moment, not the skewness)
(这是最原始的时刻，不是最阴暗的时刻)
| kurtosis =
峰度 =
===Bounded Pareto distribution===
| entropy =
| 熵 =
{{See also|Truncated distribution}}
| mgf =
2012年10月22日
{{Probability distribution
| char =
2012年10月12日
| name =Bounded Pareto
}}
}}
| type =density
| pdf_image =
The bounded (or truncated) Pareto distribution has three parameters: α, L and H. As in the standard Pareto distribution α determines the shape. L denotes the minimal value, and H denotes the maximal value.
有界的(或截断的)帕累托分布有3个参数: α，l 和 h。就像标准的帕累托分布一样，α 决定形状。L 表示最小值，h 表示最大值。
| cdf_image =
| parameters =
The probability density function is
概率密度函数
<math>L > 0</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>H > L</math> [[location parameter|location]] ([[real numbers|real]])<br />
\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha},
1-left (frac { l }{ h } right) ^ alpha } ,
<math>\alpha > 0</math> [[shape parameter|shape]] (real)
| support =<math>L \leqslant x \leqslant H</math>
where L ≤ x ≤ H, and α > 0.
where L ≤ x ≤ H, and α > 0.
| pdf =<math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
| cdf =<math>\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
| mean =
If U is uniformly distributed on (0, 1), then applying inverse-transform method
如果 u 在(0,1)上是均匀分布的，则应用逆变换方法
<math>\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1 </math><br />
<math>\frac{{H}{L}}{{H}-{L}}\ln\frac{H}{L}, \alpha=1</math>
U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}
{1-l ^ alpha x ^ {-alpha }{1-(frac { l }{ h }) ^ alpha }
| median =<math> L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}</math>
x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}
X = left (- frac { u h ^ alpha-u l ^ alpha-h ^ alpha }{ h ^ alpha ^ alpha } right) ^ {-frac {1}{ alpha }}
| mode =
| variance =
is a bounded Pareto-distributed.
是一个有界的帕累托分布。
<math>\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2</math>
<math>\frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2</math>
(this is the second raw moment, not the variance)
| skewness = <math>\frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha * (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j </math>
The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.
对称帕累托分布和零对称帕累托分布的目的是捕捉一些具有尖锐概率峰值和对称长概率尾的特殊统计分布。这两种分布是从帕累托分布中推导出来的。长概率尾通常意味着概率衰减缓慢。在很多情况下，帕累托分布的工作都很合适。但是，如果分布具有两条慢衰减尾的对称结构，帕累托不能做到这一点。然后应用对称帕累托或零对称帕累托分布。
(this is the kth raw moment, not the skewness)
| kurtosis =
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:
对称累积分布函数帕累托分布系统(CDF)定义如下:
| entropy =
| mgf =
| char =
}}
The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is
给定一个独立样本 x = (x1，x2，... ，xn) ，帕累托分布参数 α 和 xm 的似然函数为
The bounded (or truncated) Pareto distribution has three parameters: ''α'', ''L'' and ''H''. As in the standard Pareto distribution ''α'' determines the shape. ''L'' denotes the minimal value, and ''H'' denotes the maximal value.
L(\alpha, x_\mathrm{m}) = \prod_{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod_{i=1}^n \frac {1}{x_i^{\alpha+1}}.
L (alpha，x mathrm { m }) = prod { i = 1} ^ n alpha frac { x mathrm { m } ^ alpha } = alpha ^ n ^ x mathrm { m }{ n ^ { n alpha } prod { i = 1} ^ n frac {1}{ x i ^ { alpha + 1}}.
The [[probability density function]] is
Therefore, the logarithmic likelihood function is
因此，对数似然函数是
: <math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>,
\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum_{i=1} ^n \ln x_i.
Ell (alpha，x _ mathrm { m }) = n ln alpha + n alpha ln x _ mathrm { m }-(alpha + 1) sum { i = 1} ^ n ln x _ i.
where ''L'' ≤ ''x'' ≤ ''H'', and ''α'' > 0.
It can be seen that \ell(\alpha, x_\mathrm{m}) is monotonically increasing with xm, that is, the greater the value of xm, the greater the value of the likelihood function. Hence, since x ≥ xm, we conclude that
可以看出，ell (alpha，x _ mathrum { m })随 xm 单调递增，即 xm 值越大，似然函数的值越大。因此，由于 x ≥ xm，我们得出结论:
====Generating bounded Pareto random variables====
If ''U'' is [[uniform distribution (continuous)|uniformly distributed]] on (0, 1), then applying inverse-transform method <ref>http://www.cs.bgu.ac.il/~mps042/invtransnote.htm</ref>
\widehat x_\mathrm{m} = \min_i {x_i}.
广义的 x mathrm { m } = min i { x i }。
:<math>U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}</math>
To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:
为了找到 α 的估计量，我们计算相应的偏导数，并确定它在哪里为零:
:<math>x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}</math>
\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.
{ partial alpha } = frac { n }{ alpha } + n ln x _ mathrm { m }-sum _ { i = 1} ^ n ln x _ i = 0.
is a bounded Pareto-distributed.{{Citation needed|date=February 2011}}
{{Clear}}
Thus the maximum likelihood estimator for α is:
因此，α 的最大似然估计量是:
===Symmetric Pareto distribution===
\widehat \alpha = \frac{n}{\sum _i \ln (x_i/\widehat x_\mathrm{m}) }.
广义 alpha = frac { n }{ sum _ i ln (x _ i/widehat x _ mathrm { m })}。
The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.<ref name=":0">{{Cite journal|last=Huang|first=Xiao-dong|date=2004|title=A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic|journal=IEEE Transactions on Broadcasting|volume=50|issue=3|pages=323–334|doi=10.1109/TBC.2004.834013}}</ref>
The expected statistical error is:
预期的统计错误是:
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:<ref name=":0" />
\sigma = \frac {\widehat \alpha} {\sqrt n}.
1.1.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.
<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over 2b-X}) ^a & X<b \\
Malik (1970) gives the exact joint distribution of (\hat{x}_\mathrm{m},\hat\alpha). In particular, \hat{x}_\mathrm{m} and \hat\alpha are independent and \hat{x}_\mathrm{m} is Pareto with scale parameter xm and shape parameter nα, whereas \hat\alpha has an inverse-gamma distribution with shape and scale parameters n − 1 and nα, respectively.
Malik (1970)给出了(hat { x } _ mathrm { m } ，hat alpha)的精确联合分布。特别地，hat { x } _ mathrm { m }和 hat alpha 是独立的，hat { x } _ mathrm { m }是带有尺度参数 xm 和形状参数 nα 的 Pareto 分布，而 hat alpha 是带有形状参数 n-1和尺度参数 nα 的反 γ 分布。
1- \tfrac{1}{2}(\tfrac{b}{X})^a& X\geq b
\end{cases}</math>
The corresponding probability density function (PDF) is:<ref name=":0" />
Vilfredo Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income. This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population controls 80% of the wealth. However, the 80-20 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his Cours d'économie politique indicates that about 30% of the population had about 70% of the income. The probability density function (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, net worth may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:
维尔弗雷多 · 帕雷托最初用这种分配来描述个人之间的财富分配，因为它似乎很好地表明了任何社会的大部分财富被该社会中较小比例的人拥有的方式。他还用它来描述收入的分配。这种观点有时被简单地表述为帕雷托法则或“80-20法则” ，即20% 的人口控制着80% 的财富。然而，80-20法则对应的是一个特定的 α 值，事实上，帕雷托在他的《政治经济学》中关于英国所得税的数据表明，大约30% 的人口拥有大约70% 的收入。本文开头的概率密度函数图表显示，人均拥有少量财富的“概率”或人口比例相当高，然后随着财富的增加而稳步下降。(然而，帕累托分布对于低端人群的财富是不现实的。事实上，净资产甚至可能是负值。)这种分配不仅限于描述财富或收入，而且在许多情况下，在从”小”到”大”的分配中找到了平衡。下面的例子有时被看作是近似帕累托分布的:
<!-- THESE TWO SEEM TO BELONG UNDER Zipf's law RATHER THAN THE PARETO DISTRIBUTION
< ! ——这两个似乎属于 Zipf 定律，而不是帕累托分布
<math>p(x) = {ab^a \over 2(b+\left\vert x-b \right\vert)^{a+1}},X\in R</math>
This distribution has two parameters: a and b. It is symmetric by b. Then the mathematic expectation is b. When, it has variance as following:
<math>E((x-b)^2)=\int_{-\infty}^{\infty} (x-b)^2p(x)dx={2b^2 \over (a-2)(a-1) }
Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq, see also distribution fitting ]]
使用累积帕累托(洛马克斯)分布最大一天降雨量[ CumFreq，参见分布拟合]
</math>
The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:
<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over 2b-X}) ^a & X<0 \\
1- \tfrac{1}{2}(\tfrac{b}{b+X})^a& X\geq 0
\end{cases}</math>
The Pareto distribution is a continuous probability distribution. Zipf's law, also sometimes called the zeta distribution, is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if the x values (incomes) are binned into N ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining x_m so that \alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)} where H(N,\alpha-1) is the generalized harmonic number. This makes Zipf's probability density function derivable from Pareto's.
帕累托分布是一个连续的概率分布。Zipf 定律，有时也被称为 zeta 分布，是一个离散的分布，将数值分离成一个简单的排名。两者都是负指数的简单幂律，按比例调整后，它们的累积分布等于1。如果 x 值(收入)被绑定到 n 个等级中，那么每个帕累托分布中的人数遵循1/等级模式，Zipf 就可以从中得到。通过定义 x _ m 使得 αx _ mathrm { m } ^ alpha = frac {1}{ h (n，α1)}其中 h (n，α1)是广义调和数，从而将分布归一化。这使得 Zipf 的概率密度函数可以从 Pareto 衍生出来。
The corresponding PDF is:
f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)}
F (x) = frac { alpha x _ mathrm { m } ^ alpha }{ x ^ { alpha + 1}} = frac {1}{ x ^ s h (n，s)}
<math>p(x) = {ab^a \over 2(b+\left\vert x \right\vert)^{a+1}},X\in R</math>
where s = \alpha-1 and x is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has f(x) probability of ranking x.
其中 s = alpha-1和 x 是一个整数，表示从1到 n 的等级，其中 n 是最高的收入等级。因此，从一个人口(或语言、互联网或国家)中随机选择的一个人(或单词、网站链接或城市)具有 f (x)排名概率 x。
This distribution is symmetric by zero. Parameter is related to the decay rate of probability and represents peak magnitude of probability.<ref name=":0" />
The "80-20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is \alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown to be mathematically equivalent:
根据“80-20法则” ，20% 的人收入占总收入的80% ，20% 最富裕的20% 收入占总收入的80% ，以此类推，当帕累托指数为 alpha = log 45 = cfrac { log 10}5 log {10}4}大约1.161时，恰好适用。这个结果可以从下面给出的洛伦兹曲线公式推导出来。此外，经证明以下几点在数学上是等价的:
===[[Multivariate Pareto distribution]]===
The univariate Pareto distribution has been extended to a [[multivariate distribution|multivariate]] Pareto distribution.<ref>{{cite journal
|last1=Rootzén|first1=Holger |last2=Tajvidi|first2=Nader
1-\frac{1}{\alpha}=\frac{\ln(1-p^n)}{\ln(1-(1-p)^n)}
1-frac {1}{ alpha } = frac { ln (1-p ^ n)}{ ln (1-(1-p) ^ n)}
|title=Multivariate generalized Pareto distributions |journal=Bernoulli|volume=12|year=2006|number=5 |pages=917–30
|doi=10.3150/bj/1161614952
This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
这不仅适用于收入，也适用于财富，或任何其他可以由这种分配模式。
|ref=harv |citeseerx=10.1.1.145.2991}}</ref>
This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.
这排除了0 < α ≤1的帕累托分布，如上所述，这种分布具有无限的期望值，因此不能合理地模拟收入分配。
==Statistical Inference==
===Estimation of parameters===
Price's square root law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that \alpha=1. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.
普莱斯的平方根定律有时被作为帕累托分布的性质或类似的性质提供。然而，该定律只适用于 alpha = 1的情况。请注意，在这种情况下，财富的总量和期望值是没有定义的，这个规则只是渐近地适用于随机样本。上面提到的扩展的帕雷托法则是一个更普遍的规则。
The [[likelihood function]] for the Pareto distribution parameters ''α'' and ''x''<sub>m</sub>, given an independent [[sample (statistics)|sample]] ''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>''), is
: <math>L(\alpha, x_\mathrm{m}) = \prod_{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod_{i=1}^n \frac {1}{x_i^{\alpha+1}}.</math>
Lorenz curves for a number of Pareto distributions. The case α = ∞ corresponds to perfectly equal distribution (G = 0) and the α = 1 line corresponds to complete inequality (G = 1)
一类 Pareto 分布的 Lorenz 曲线。情形 α = ∞对应完全等分布(g = 0) ，α = 1线对应完全不等式(g = 1)
Therefore, the logarithmic likelihood function is
The Lorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve L(F) is written in terms of the PDF f or the CDF F as
洛伦兹曲线常被用来描述收入和财富的分配。对于任何分布，洛伦兹曲线 l (f)用 PDF 表示，或用 CDF 表示
: <math>\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum_{i=1} ^n \ln x_i.</math>
L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}
L (f) = frac { int _ { x _ mathrm { m } ^ { x (f)} xf (x) ，dx }{ int _ { x _ mathrm { m } ^ infty xf (x) ，dx } = frac { int _ 0 ^ f x (f’) ，dF’}{ int _ 0 ^ 1 x (f’) ，dF’}
It can be seen that <math>\ell(\alpha, x_\mathrm{m})</math> is monotonically increasing with ''x''<sub>m</sub>, that is, the greater the value of ''x''<sub>m</sub>, the greater the value of the likelihood function. Hence, since ''x'' ≥ ''x''<sub>m</sub>, we conclude that
where x(F) is the inverse of the CDF. For the Pareto distribution,
其中 x (f)是 CDF 的倒数。对于帕累托分布来说,
: <math>\widehat x_\mathrm{m} = \min_i {x_i}.</math>
x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}}
x (f) = frac { x _ mathrm { m }{(1-F) ^ { frac {1}{ alpha }}}}
To find the [[estimator]] for ''α'', we compute the corresponding partial derivative and determine where it is zero:
and the Lorenz curve is calculated to be
计算出洛伦兹曲线为
: <math>\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.</math>
L(F) = 1-(1-F)^{1-\frac{1}{\alpha}},
1-(1-F) ^ {1-frac {1}{ alpha } ,
Thus the [[maximum likelihood]] estimator for ''α'' is:
For 0<\alpha\le 1 the denominator is infinite, yielding L=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
对于0 < alpha le 1，分母是无穷大，产生 l = 0。右图显示了一些帕累托分布的洛伦兹曲线的例子。
: <math>\widehat \alpha = \frac{n}{\sum _i \ln (x_i/\widehat x_\mathrm{m}) }.</math>
According to Oxfam (2016) the richest 62 people have as much wealth as the poorest half of the world's population. We can estimate the Pareto index that would apply to this situation. Letting ε equal 62/(7\times 10^9) we have:
根据牛津饥荒救济委员会(Oxfam)(2016)的数据，最富有的62个人拥有的财富相当于世界上最贫穷的一半人口的财富。我们可以估计适用于这种情况的帕累托指数。让 ε 等于62/(7乘以10 ^ 9)我们有:
L(1/2)=1-L(1-\varepsilon)
L (1/2) = 1-L (1-varepsilon)
The expected statistical error is:<ref>{{cite journal |author=M. E. J. Newman |year=2005 |title=Power laws, Pareto distributions and Zipf's law |journal=[[Contemporary Physics]] |volume=46 |issue=5 |pages=323–51| arxiv=cond-mat/0412004 |doi=10.1080/00107510500052444 |bibcode=2005ConPh..46..323N|s2cid=202719165 }}</ref>
or
或
1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}}
1-(1/2) ^ {1-frac {1}{ alpha }}} = varepsilon ^ {1-frac {1}{ alpha }}}
: <math>\sigma = \frac {\widehat \alpha} {\sqrt n}. </math>
<!--:\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(1-\frac{1}{\alpha})\ln\varepsilon
< ! -- : ln (1-(1/2) ^ {1-frac {1}{ alpha }}) = (1-frac {1}{ alpha }) ln varepsilon
\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(\ln\varepsilon/\ln 2)(1-\frac{1}{\alpha})\ln 2
Ln (1-(1/2) ^ {1-frac {1}{ alpha }}) = (ln varepsilon/ln 2)(1-frac {1}{ alpha }) ln 2
Malik (1970)<ref>{{cite journal |author=H. J. Malik |year=1970 |title=Estimation of the Parameters of the Pareto Distribution |journal=Metrika |volume=15|pages=126–132 |doi=10.1007/BF02613565 |s2cid=124007966 }}</ref> gives the exact joint distribution of <math>(\hat{x}_\mathrm{m},\hat\alpha)</math>. In particular, <math>\hat{x}_\mathrm{m}</math> and <math>\hat\alpha</math> are [[Independence (probability theory)|independent]] and <math>\hat{x}_\mathrm{m}</math> is Pareto with scale parameter ''x''<sub>m</sub> and shape parameter ''nα'', whereas <math>\hat\alpha</math> has an [[inverse-gamma distribution]] with shape and scale parameters ''n'' − 1 and ''nα'', respectively.
\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})
Ln (1-(1/2) ^ {1-frac {1}{ alpha }}) = -- lrb- ln varepsilon/ln 2) ln ((1/2) ^ {1-frac {1}{ alpha })
\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx(\ln\varepsilon/\ln 2)(1-(1/2)^{1-\frac{1}{\alpha}})
Ln (1-(1/2) ^ {1-frac {1}{ alpha }}) approx (ln varepsilon/ln 2)(1-(1/2) ^ {1-frac {1}{ alpha })
==Occurrence and applications==
-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\exp(-\ln(1-(1/2)^{1-\frac{1}{\alpha}}))\approx -\ln\varepsilon/\ln 2
- ln (1-(1/2) ^ {1-frac {1}{ alpha }}) exp (- ln (1-(1/2) ^ {1-frac {1}{ alpha }})) approx-ln varepsilon/ln 2
===General===
-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)
- ln (1-(1/2) ^ {1-frac {1}{ alpha }}) approx w (- ln varepsilon/ln 2)
[[Vilfredo Pareto]] originally used this distribution to describe the [[Distribution of wealth|allocation of wealth]] among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.<ref>Pareto, Vilfredo, ''Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino'', Librairie Droz, Geneva, 1964, pp. 299–345.</ref> This idea is sometimes expressed more simply as the [[Pareto principle]] or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.<ref>For a two-quantile population, where approximately 18% of the population owns 82% of the wealth, the [[Theil index]] takes the value 1.</ref> However, the 80-20 rule corresponds to a particular value of ''α'', and in fact, Pareto's data on British income taxes in his ''Cours d'économie politique'' indicates that about 30% of the population had about 70% of the income.{{citation needed|date=May 2019}} The [[probability density function]] (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, [[net worth]] may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:
where W is the Lambert W function. So
在那里 w 是朗伯W函数。所以
<!-- THESE TWO SEEM TO BELONG UNDER [[Zipf's law]] RATHER THAN THE PARETO DISTRIBUTION
(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))
(1/2) ^ {1-frac {1}{ alpha }} approx 1-exp (- w (- ln varepsilon/ln 2))
* Frequencies of words in longer texts (a few words are used often, lots of words are used infrequently)
{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2
{1-frac {1}{ alpha }}} approx-ln (1-exp (- w (- ln varepsilon/ln 2)))/ln 2
* Frequencies of [[Given name#Popularity distribution of given names|given names]] -->
\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)
(1-exp (- w (- ln varepsilon/ln 2))/ln 2)
* The sizes of human settlements (few cities, many hamlets/villages)<ref name="Reed">{{cite journal |citeseerx=10.1.1.70.4555 |first=William J. |last=Reed |title=The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions |journal=Communications in Statistics – Theory and Methods |volume=33 |issue=8 |pages=1733–53 |year=2004 |doi=10.1081/sta-120037438|s2cid=13906086 |display-authors=etal}}</ref><ref name="Reed2002">{{cite journal |first=William J. |last=Reed |title=On the rank‐size distribution for human settlements |journal=Journal of Regional Science |volume=42 |issue=1 |pages=1–17 |year=2002 |doi=10.1111/1467-9787.00247|s2cid=154285730 }}</ref>
-->The solution is that α equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.
解决方案是，α 大约等于1.15，这两个群体各拥有大约9% 的财富。但实际上，世界上最贫穷的69% 的成年人只拥有大约3% 的财富。
* File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)<ref name ="Reed" />
* [[Hard disk drive]] error rates<ref>{{cite journal |title=Understanding latent sector error and how to protect against them |url=http://www.usenix.org/event/fast10/tech/full_papers/schroeder.pdf |first1=Bianca |last1=Schroeder |first2=Sotirios |last2=Damouras |first3=Phillipa |last3=Gill |journal=8th Usenix Conference on File and Storage Technologies (FAST 2010)| date=2010-02-24 |accessdate=2010-09-10 |quote=We experimented with 5 different distributions (Geometric,Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ<sup>2</sup> statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.}}</ref>
The Gini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (α = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for \alpha\ge 1) to be
基尼系数是衡量洛伦兹曲线与等分布线之间的偏差，等分布线是一条连接[0,0]和[1,1]的线，右边的洛伦兹曲线用黑色(α = ∞)表示。具体来说，基尼系数是洛伦兹曲线和等分布线之间面积的两倍。然后计算出基尼系数帕累托分布的平均值(对于 alpha ge 1)为
* Clusters of [[Bose–Einstein condensate]] near [[absolute zero]]<ref name="Simon">{{cite journal|first2=Herbert A.|last2=Simon|author=Yuji Ijiri |title=Some Distributions Associated with Bose–Einstein Statistics|journal=Proc. Natl. Acad. Sci. USA|date=May 1975|volume=72|issue=5|pages=1654–57|pmc=432601|pmid=16578724|doi=10.1073/pnas.72.5.1654|bibcode=1975PNAS...72.1654I}}</ref>
[[File:FitParetoDistr.tif|thumb|250px|Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] ]]
G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1}
1-2 left (int _ 0 ^ 1L (f) ，dF right) = frac {1}{2 alpha-1}
* The values of [[oil reserves]] in oil fields (a few [[Giant oil and gas fields|large fields]], many [[Stripper well|small fields]])<ref name ="Reed" />
* The length distribution in jobs assigned to supercomputers (a few large ones, many small ones)<ref>{{Cite journal|last1=Harchol-Balter|first1=Mor|author1-link=Mor Harchol-Balter|last2=Downey|first2=Allen|date=August 1997|title=Exploiting Process Lifetime Distributions for Dynamic Load Balancing|url=https://users.soe.ucsc.edu/~scott/courses/Fall11/221/Papers/Sync/harcholbalter-tocs97.pdf|journal=ACM Transactions on Computer Systems|volume=15|issue=3|pages=253–258|doi=10.1145/263326.263344|s2cid=52861447}}</ref>
(see Aaberge 2005).
(见 Aaberge 2005)。
* The standardized price returns on individual stocks <ref name="Reed" />
* Sizes of sand particles <ref name ="Reed" />
* The size of meteorites
* Male dating success on Tinder [80% of females compete for the 20% most attractive males] <ref name="Medium.com">[https://medium.com/p/2ddf370a6e9a]</ref>
Random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0, 1], the variate T given by
利用逆变换采样生成随机样本。给定从单位区间上的均匀分布得到的随机变量 u (0,1] ，给出了 t 的变量 t
* Severity of large [[casualty (person)|casualty]] losses for certain lines of business such as general liability, commercial auto, and workers compensation.<ref>Kleiber and Kotz (2003): p. 94.</ref><ref>{{cite journal |last1=Seal |first1=H. |year=1980 |title=Survival probabilities based on Pareto claim distributions |journal=ASTIN Bulletin |volume=11 |pages=61–71|doi=10.1017/S0515036100006620 |doi-access=free }}</ref>
* Amount of time a user on [[Steam (software)|Steam]] will spend playing different games. (Some games get played a lot, but most get played almost never.) [https://docs.google.com/spreadsheets/d/1BDv2W4IsgxiAhhUznTbMSfRtcLia320Zq1HxzwhKao0/edit#gid=0]
T=\frac{x_\mathrm{m}}{U^{1/\alpha}}
1/alpha }}
* In [[hydrology]] the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.<ref>CumFreq, software for cumulative frequency analysis and probability distribution fitting [https://www.waterlog.info/cumfreq.htm]</ref> The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].
is Pareto-distributed. If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).
是帕累托分布的。如果 u 在[0,1]上是均匀分布的，则它可以与(1-u)交换。
===Relation to Zipf's law===
The Pareto distribution is a continuous probability distribution. [[Zipf's law]], also sometimes called the [[zeta distribution]], is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if the <math>x</math> values (incomes) are binned into <math>N</math> ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining <math>x_m</math> so that <math>\alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)}</math> where <math>H(N,\alpha-1)</math> is the [[Harmonic number#Generalized harmonic numbers|generalized harmonic number]]. This makes Zipf's probability density function derivable from Pareto's.
: <math>f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)}</math>
where <math>s = \alpha-1</math> and <math>x</math> is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has <math>f(x)</math> probability of ranking <math>x</math>.
===Relation to the "Pareto principle"===
The "[[80-20 law]]", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is <math>\alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161</math>. This result can be derived from the [[Lorenz curve]] formula given below. Moreover, the following have been shown<ref>{{cite journal |last1=Hardy |first1=Michael |year=2010 |title=Pareto's Law |journal=[[Mathematical Intelligencer]] |volume=32 |issue=3 |pages=38–43 |doi=10.1007/s00283-010-9159-2|s2cid=121797873 }}</ref> to be mathematically equivalent:
* Income is distributed according to a Pareto distribution with index ''α'' > 1.
* There is some number 0 ≤ ''p'' ≤ 1/2 such that 100''p'' % of all people receive 100(1 − ''p'')% of all income, and similarly for every real (not necessarily integer) ''n'' > 0, 100''p<sup>n</sup>'' % of all people receive 100(1 − ''p'')<sup>''n''</sup> percentage of all income. ''α'' and ''p'' are related by
:: <math>1-\frac{1}{\alpha}=\frac{\ln(1-p^n)}{\ln(1-(1-p)^n)}</math>
This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
This excludes Pareto distributions in which 0 < ''α'' ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.
===Relation to Price's law===
[[Derek J. de Solla Price#Scientific contributions|Price's square root law]] is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that <math>\alpha=1</math>. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.
===Lorenz curve and Gini coefficient===
| title=Ecrits sur la courbe de la répartition de la richesse
| title=Ecrits sur la courbe de la répartition de la richesse
[[File:ParetoLorenzSVG.svg|thumb|325px|Lorenz curves for a number of Pareto distributions. The case ''α'' = ∞ corresponds to perfectly equal distribution (''G'' = 0) and the ''α'' = 1 line corresponds to complete inequality (''G'' = 1)]]
| first=Vilfredo
第一名: 维尔弗雷多
| last=Pareto
最后 = 帕累托
The [[Lorenz curve]] is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve ''L''(''F'') is written in terms of the PDF ''f'' or the CDF ''F'' as
| editor=Librairie Droz
编辑: Librairie Droz
| year=1965
1965年
:<math>L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}</math>
| pages=48
48
| series=Œuvres complètes : T. III
| series=Œuvres complètes : T. III
where ''x''(''F'') is the inverse of the CDF. For the Pareto distribution,
| isbn=9782600040211}}
9782600040211}
:<math>x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}}</math>
and the Lorenz curve is calculated to be
| first=Vilfredo
第一名: 维尔弗雷多
| last=Pareto
最后 = 帕累托
:<math>L(F) = 1-(1-F)^{1-\frac{1}{\alpha}},</math>
| year=1896
1896年
| title=Cours d'économie politique
| title=Cours d'économie politique
For <math>0<\alpha\le 1</math> the denominator is infinite, yielding ''L''=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
| doi=10.1177/000271629700900314| s2cid=143528002
10.1177/000271629700900314 | s2cid = 143528002
}}
}}
According to [[Oxfam]] (2016) the richest 62 people have as much wealth as the poorest half of the world's population.<ref>{{cite web|title=62 people own the same as half the world, reveals Oxfam Davos report|url=https://www.oxfam.org/en/pressroom/pressreleases/2016-01-18/62-people-own-same-half-world-reveals-oxfam-davos-report|publisher=Oxfam|date=Jan 2016}}</ref> We can estimate the Pareto index that would apply to this situation. Letting ε equal <math>62/(7\times 10^9)</math> we have:
:<math>L(1/2)=1-L(1-\varepsilon)</math>
or
:<math>1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}}</math>
<!--:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(1-\frac{1}{\alpha})\ln\varepsilon</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(\ln\varepsilon/\ln 2)(1-\frac{1}{\alpha})\ln 2</math>
| url=http://www3.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf
Http://www3.unisi.it/eventi/ginilorenz05/25%20may%20paper/paper_aaberge.pdf
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})</math>
| title=Gini's Nuclear Family
| title = Gini 的核心家庭
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx(\ln\varepsilon/\ln 2)(1-(1/2)^{1-\frac{1}{\alpha}})</math>
| first=Rolf
第一个 = Rolf
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\exp(-\ln(1-(1/2)^{1-\frac{1}{\alpha}}))\approx -\ln\varepsilon/\ln 2</math>
| last=Aabergé
| last=Aabergé
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)</math>
| conference=International Conference to Honor Two Eminent Social Scientists
两位杰出社会科学家荣誉国际会议
where ''W'' is the [[Lambert W function]]. So
| date=May 2005}}
2005年5月
:<math>(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))</math>
:<math>{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2</math>
:<math>\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)</math>
| url=https://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
Https://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
-->The solution is that ''α'' equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.<ref>{{cite web|title=Global Wealth Report 2013|url=https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|publisher=Credit Suisse|page=22|date=Oct 2013|access-date=2016-01-24|archive-url=https://web.archive.org/web/20150214155424/https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|archive-date=2015-02-14|url-status=dead}}</ref>
| title=Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes
| title = 万维网流量的自相似性: 证据和可能的原因
| first1=Mark E.
1 = Mark e.
The [[Gini coefficient]] is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (''α'' = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for <math>\alpha\ge 1</math>) to be
| last1=Crovella
1 = Crovella
| author-link1=Mark Crovella
1 = Mark Crovella
:<math>G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1}</math>
| first2=Azer
2 = Azer
| last2=Bestavros
2 = Bestavros
(see Aaberge 2005).
| conference=IEEE/ACM Transactions on Networking
会议 = IEEE/ACM网络学报
| volume=5
5
==Computational methods==
| number=6
6
===Random sample generation===
| pages=835–846
| 页数 = 835-846
Random samples can be generated using [[inverse transform sampling]]. Given a random variate ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] on the unit interval (0, 1], the variate ''T'' given by
| date=December 1997}}
| date = December 1997}
:<math>T=\frac{x_\mathrm{m}}{U^{1/\alpha}}</math>
is Pareto-distributed.<ref>{{cite book |last=Tanizaki |first=Hisashi |title=Computational Methods in Statistics and Econometrics |year=2004 |page=133 |publisher=CRC Press |url=https://books.google.com/books?id=pOGAUcn13fMC|isbn=9780824750886 }}</ref> If ''U'' is uniformly distributed on [0, 1), it can be exchanged with (1 − ''U'').
==See also==
* [[Bradford's law]]
* [[Gutenberg–Richter law]]
* [[Matthew effect]]
Category:Actuarial science
类别: 精算
* [[Pareto analysis]]
Category:Continuous distributions
类别: 连续分布
* [[Pareto efficiency]]
Category:Power laws
分类: 权力法则
* [[Pareto interpolation]]
Category:Probability distributions with non-finite variance
范畴: 非有限方差的概率分布
* [[Power law#Power-law probability distributions|Power law probability distributions]]
Category:Exponential family distributions
类别: 指数族分布
* [[Sturgeon's law]]
Category:Vilfredo Pareto
类别: Vilfredo Pareto
<noinclude>
<small>This page was moved from [[wikipedia:en:Pareto distribution]]. Its edit history can be viewed at [[Pareto分布/edithistory]]</small></noinclude>
[[Category:待整理页面]]
{{Probability distribution
{{Probability distribution
{概率分布
| name =Pareto Type I
| name =Pareto Type I
| name = Pareto Type i
| type =density
| type =density
类型 = 密度
| pdf_image =[[File:Probability density function of Pareto distribution.svg|325px|Pareto Type I probability density functions for various ''α'']]<br />Pareto Type I probability density functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math> As <math>\alpha \rightarrow \infty,</math> the distribution approaches <math>\delta(x - x_\mathrm{m}),</math> where <math>\delta</math> is the [[Dirac delta function]].
| pdf_image =Pareto Type I probability density functions for various α<br />Pareto Type I probability density functions for various \alpha with x_\mathrm{m} = 1. As \alpha \rightarrow \infty, the distribution approaches \delta(x - x_\mathrm{m}), where \delta is the Dirac delta function.
| pdf _ image = 各种 α < br/> Pareto i 型概率密度函数的 Pareto i 型概率密度函数。在 α 向右下方，分布趋近于 δ (x-x _ mathrm { m }) ，其中 δ 是狄拉克δ函数。
| cdf_image =[[File:Cumulative distribution function of Pareto distribution.svg|325px|Pareto Type I cumulative distribution functions for various ''α'']]<br />Pareto Type I cumulative distribution functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math>
| cdf_image =Pareto Type I cumulative distribution functions for various α<br />Pareto Type I cumulative distribution functions for various \alpha with x_\mathrm{m} = 1.
不同 α < br/> Pareto i 型累积分布函数的 Pareto i 型累积分布函数。
| parameters =<math>x_\mathrm{m} > 0</math> [[scale parameter|scale]] ([[real number|real]])<br /><math>\alpha > 0</math> [[shape parameter|shape]] (real)
| parameters =x_\mathrm{m} > 0 scale (real)<br />\alpha > 0 shape (real)
0 scale (real) < br/> alpha > 0 shape (real)
| support =<math>x \in [x_\mathrm{m}, \infty)</math>
| support =x \in [x_\mathrm{m}, \infty)
| support = x in [ x _ mathrm { m } ，infty)
| pdf =<math>\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}</math>
| pdf =\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}
1}{ x ^ { alpha + 1}}
| cdf =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha</math>
| cdf =1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha
1-left (frac { x _ mathrm { m }{ x } right) ^ alpha
| mean =<math>\begin{cases}
| mean =<math>\begin{cases}
开始{ cases }
\infty & \text{for }\alpha\le 1 \\
\infty & \text{for }\alpha\le 1 \\
1.1
\dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
\dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
1 & text { for } alpha > 1
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
| median =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math>
| median =x_\mathrm{m} \sqrt[\alpha]{2}
中位数 = x _ mathrm { m } sqrt [ alpha ]{2}
| mode =<math>x_\mathrm{m}</math>
| mode =x_\mathrm{m}
2009年10月11日
| variance =<math>\begin{cases}
| variance =<math>\begin{cases}
| 方差 = < math > begin { cases }
\infty & \text{for }\alpha\le 2 \\
\infty & \text{for }\alpha\le 2 \\
2.1.1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2
\dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2
\dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2
1) ^ 2(alpha-2)} & text { for } alpha > 2
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
| skewness =<math>\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3</math>
| skewness =\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3
| skewness = frac {2(1 + alpha)}{ alpha-3} sqrt { frac { alpha-2}{ alpha }}} text { for } alpha > 3
| kurtosis =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4</math>
| kurtosis =\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4
| 峭度 = frac {6(alpha ^ 3 + alpha ^ 2-6 alpha-2)}{ alpha (alpha-3)(alpha-4)} text { for } alpha > 4
| entropy =<math>\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right) </math>
| entropy =\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right)
| 熵 = log left (left (left (frac { x _ mathrm { m }{ alpha }右) ，e ^ {1 + tfrac {1}{ alpha }右)
| mgf =<math>\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0</math>
| mgf =\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0
| mgf = alpha (- x _ mathrm { m } t) ^ alpha Gamma (- alpha,-x _ mathrm { m } t) text { for } t < 0
| char =<math>\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)</math>
| char =\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)
| char = alpha (- ix _ mathrm { m } t) ^ alpha Gamma (- alpha,-ix _ mathrm { m } t)
| fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
| fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
| fisher = < math > mathcal { i }(x _ mathrm { m } ，alpha) = begin { bmatrix }
\dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\
\dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\
2} &-dfrac {1}{ x mathrm { m }
-\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2}
-\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2}
- dfrac {1}{ x _ mathrm { m } & dfrac {1}{ alpha ^ 2}
\end{bmatrix}</math>
\end{bmatrix}</math>
结束{ bmatrix } </math >
Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
右: < math > mathcal { i }(x _ mathrm { m } ，alpha) = begin { bmatrix }
\dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
\dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
2} & 0
0 & \dfrac{1}{\alpha^2}
0 & \dfrac{1}{\alpha^2}
0 & dfrac {1}{ alpha ^ 2}
\end{bmatrix}</math>
\end{bmatrix}</math>
结束{ bmatrix } </math >
}}
}}
}}
The '''Pareto distribution''', named after the Italian [[civil engineer]], [[economist]], and sociologist [[Vilfredo Pareto]],<ref>{{Cite journal|last=Amoroso|first=Luigi|date=1938|title=VILFREDO PARETO|url=|journal=Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest|volume=6|pages=|via=}}</ref> is a [[power-law]] [[probability distribution]] that is used in description of [[social sciences|social]], [[scientific]], [[geophysical]], [[actuarial science|actuarial]], and many other types of observable phenomena. Originally applied to describing the [[distribution of wealth]] in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population,<ref>{{Cite journal|last=Pareto|first=Vilfredo|date=1898|title=Cours d'economie politique|url=|journal=Journal of Political Economy|volume=6|pages=|via=}}</ref> the Pareto distribution has colloquially become known and referred to as the [[Pareto principle]], or "80-20 rule", and is sometimes called the "[[Matthew principle]]".<!-- can we find a better reference than https://youtu.be/5WX9UEYZsR8 at 2'10". --> This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, <math>\alpha</math> (''α'' = log<sub>4</sub>5 ≈ 1.16). While <math>\alpha</math> is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena<ref>{{Cite journal|last=VAN MONTFORT|first=M.A.J.|date=1986|title=The Generalized Pareto distribution applied to rainfall depths|journal=Hydrological Sciences Journal|volume=31|issue=2|pages=151–162|doi=10.1080/02626668609491037|doi-access=free}}</ref> and human activities.<ref>{{Cite journal|last=Oancea|first=Bogdan|date=2017|title=Income inequality in Romania: The exponential-Pareto distribution|journal=Physica A: Statistical Mechanics and Its Applications|volume=469|pages=486–498|doi=10.1016/j.physa.2016.11.094|bibcode=2017PhyA..469..486O}}</ref>
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. Originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population, the Pareto distribution has colloquially become known and referred to as the Pareto principle, or "80-20 rule", and is sometimes called the "Matthew principle".<!-- can we find a better reference than https://youtu.be/5WX9UEYZsR8 at 2'10". --> This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, \alpha (α = log45 ≈ 1.16). While \alpha is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena and human activities.
以意大利土木工程师、经济学家和社会学家 Vilfredo Pareto 命名的帕累托分布概率分布，是一种用于描述社会、科学、地球物理、保险精算和许多其他类型的可观测现象的幂定律。最初用于描述一个社会中的财富分配，符合大部分财富由一小部分人口持有的趋势，帕累托分布被通俗地称为帕雷托法则法则，或“80-20法则” ，有时也被称为“马太原则”。<！——我们能找到比2英尺10英寸的 https://youtu.be/5wx9ueyzsr8更好的参考吗。例如，这条规则规定，一个社会80% 的财富掌握在20% 的人口手中。然而，我们不应该把帕累托分布与帕雷托法则混为一谈，因为前者只对一个特定的幂值产生这个结果，即 α = log45≈1.16。虽然 α 值是一个参数，但经验观察发现80-20分布适用于各种情况，包括自然现象和人类活动。
==Definitions==
If ''X'' is a [[random variable]] with a Pareto (Type I) distribution,<ref name=arnold>{{cite book |author=Barry C. Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn= 978-0-89974-012-6|ref=harv}}</ref> then the probability that ''X'' is greater than some number ''x'', i.e. the [[survival function]] (also called tail function), is given by
If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by
如果 x 是一个带有 Pareto (Type i)分布的随机变量，那么 x 大于某个数 x 的概率，即。生存函数(也称为尾部函数)由
:<math>\overline{F}(x) = \Pr(X>x) = \begin{cases}
<math>\overline{F}(x) = \Pr(X>x) = \begin{cases}
< math > overline { f }(x) = Pr (x > x) = begin { cases }
\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
左(frac { x _ mathrm { m }{ x }右) ^ alpha & x ge x _ mathrm { m } ,
1 & x < x_\mathrm{m},
1 & x < x_\mathrm{m},
1 & x < x mathrm { m } ,
\end{cases}
\end{cases}
结束{ cases }
</math>
</math>
数学
where ''x''<sub>m</sub> is the (necessarily positive) minimum possible value of ''X'', and ''α'' is a positive parameter. The Pareto Type I distribution is characterized by a [[scale parameter]] ''x''<sub>m</sub> and a [[shape parameter]] ''α'', which is known as the ''tail index''. When this distribution is used to model the distribution of wealth, then the parameter ''α'' is called the [[Pareto index]].
where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
其中 xm 是 x 的(必然是正的)最小可能值，α 是一个正参数。帕累托 i 型分布的拥有属性是尺度参数 xm 和形状参数 α，即众所周知的尾部指数。当这个分布被用来模拟财富的分布时，参数 α 被称为帕累托指数。
===Cumulative distribution function===
From the definition, the [[cumulative distribution function]] of a Pareto random variable with parameters ''α'' and ''x''<sub>m</sub> is
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is
根据定义，带有参数 α 和 xm 的 Pareto 随机变量的累积分布函数是
:<math>F_X(x) = \begin{cases}
<math>F_X(x) = \begin{cases}
< math > f _ x (x) = begin { cases }
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
1-left (frac { x _ mathrm { m }{ x } right) ^ alpha & x ge x _ mathrm { m } ,
0 & x < x_\mathrm{m}.
0 & x < x_\mathrm{m}.
0 & x < x mathrm { m }.
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
===Probability density function===
It follows (by [[Derivative|differentiation]]) that the [[probability density function]] is
It follows (by differentiation) that the probability density function is
由此可以得出结论: 概率密度函数
:<math>f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases} </math>
f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases}
F _ x (x) = begin { cases } frac { alpha x mathrm { m } ^ alpha }{ x ^ alpha + 1} & x ge x mathrm { m } ，0 & x < x mathrm { m }.结束{ cases }
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes [[asymptotically]]. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a [[log-log plot]], the distribution is represented by a straight line.
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a log-log plot, the distribution is represented by a straight line.
在直线轴上绘制时，分布曲线呈现常见的 j 形曲线，渐近地接近每个正交轴。曲线的所有部分都是自相似的(取决于适当的比例因子)。在对数对数图中绘制时，分布由一条直线表示。
==Properties==
===Moments and characteristic function===
* The [[expected value]] of a [[random variable]] following a Pareto distribution is
:
:: <math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
<math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
(x) = begin { cases } infty & alpha le 1,
\frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha>1.
\frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha>1.
1.1.1.1.1.2.1.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
* The [[variance]] of a [[random variable]] following a Pareto distribution is
:: <math>\operatorname{Var}(X)= \begin{cases}
<math>\operatorname{Var}(X)= \begin{cases}
{ Var }(x) = begin { cases }
\infty & \alpha\in(1,2], \\
\infty & \alpha\in(1,2], \\
(1,2)中的 infty & alpha
\left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha>2.
\left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha>2.
左(frac { x _ mathrm { m }{ alpha-1}右) ^ 2 frac { alpha-2} & alpha > 2。
\end{cases}</math>
\end{cases}</math>
结束{ cases } </math >
: (If ''α'' ≤ 1, the variance does not exist.)
(If α ≤ 1, the variance does not exist.)
(如果 α ≤1，方差不存在.)
* The raw [[moment (mathematics)|moments]] are
:: <math>\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}</math>
\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}
N’ = begin { cases } infty & alpha le n，frac { alpha x mathrm { m } ^ n }{ alpha-n } & alpha > n。结束{ cases }
* The [[Moment-generating function|moment generating function]] is only defined for non-positive values ''t'' ≤ 0 as
::<math>M\left(t;\alpha,x_\mathrm{m}\right) = \operatorname{E} \left [e^{tX} \right ] = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t)</math>
M\left(t;\alpha,x_\mathrm{m}\right) = \operatorname{E} \left [e^{tX} \right ] = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t)
M left (t; alpha，x _ mathrm { m } right) = operatorname { e } left [ e ^ { tX } right ] = alpha (- x _ mathrm { m } t) ^ alpha Gamma (- alpha,-x _ mathrm { m } t)
::<math>M\left(0,\alpha,x_\mathrm{m}\right)=1.</math>
M\left(0,\alpha,x_\mathrm{m}\right)=1.
M 左(0，alpha，x _ mathrm { m }右) = 1。
* The [[Characteristic function (probability theory)|characteristic function]] is given by
:: <math>\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),</math>
\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),
Varphi (t; alpha，x _ mathrm { m }) = alpha (- ix _ mathrm { m } t) ^ alpha Gamma (- alpha,-ix _ mathrm { m } t) ,
: where Γ(''a'', ''x'') is the [[incomplete gamma function]].
where Γ(a, x) is the incomplete gamma function.
其中 γ (a，x)是不完全Γ函数。
The parameters may be solved using the [[method of moments]].<!-- :
The parameters may be solved using the method of moments.<!-- :
参数可以用矩量法求解。 < ! -- :
alpha = 1 + sqr(1 + mean ^ 2 / var)
alpha = 1 + sqr(1 + mean ^ 2 / var)
Alpha = 1 + sqr (1 + mean ^ 2/var)
beta = mean * Sqr(mean ^ 2 + var) / (Sqr(var) + Sqr(mean ^ 2 + var)) -->
beta = mean * Sqr(mean ^ 2 + var) / (Sqr(var) + Sqr(mean ^ 2 + var)) -->
Beta = mean * Sqr (mean ^ 2 + var)/(Sqr (var) + Sqr (mean ^ 2 + var)) -- >
<ref>S. Hussain, S.H. Bhatti (2018).
[https://www.researchgate.net/publication/322758024_Parameter_estimation_of_Pareto_distribution_Some_modified_moment_estimators Parameter estimation of Pareto distribution: Some modified moment estimators]. ''Maejo International Journal of Science and Technology'' 12(1):11-27</ref>
===Conditional distributions===
The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number x_1 exceeding x_\text{m}, is a Pareto distribution with the same Pareto index \alpha but with minimum x_1 instead of x_\text{m}.
一个帕雷托分布的随机变量的条件概率分布，假设它大于或等于一个特定的数字 x 1超过 x _ text { m } ，是一个具有相同帕雷托指数 α 但最小 x _ 1而不是 x _ text { m }的帕累托分布。
The [[conditional probability distribution]] of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number <math>x_1</math> exceeding <math>x_\text{m}</math>, is a Pareto distribution with the same Pareto index <math>\alpha</math> but with minimum <math>x_1</math> instead of <math>x_\text{m}</math>.
===A characterization theorem===
Suppose X_1, X_2, X_3, \dotsc are independent identically distributed random variables whose probability distribution is supported on the interval [x_\text{m},\infty) for some x_\text{m}>0. Suppose that for all n, the two random variables \min\{X_1,\dotsc,X_n\} and (X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\} are independent. Then the common distribution is a Pareto distribution.
假设 x _ 1，x _ 2，x _ 3，dotsc 是独立同分布的随机变量，对于某些 x _ text { m } > 0，其概率分布在区间[ x _ text { m } ，infty ]上是支持的。假设对于所有 n，两个随机变量 min { x1，dotsc，xn }和(x1 + dotsb + xn)/min { x1，dotsc，xn }是独立的。那么公共分配就是帕累托分布。
Suppose <math>X_1, X_2, X_3, \dotsc</math> are [[independent identically distributed]] [[random variable]]s whose probability distribution is supported on the interval <math>[x_\text{m},\infty)</math> for some <math>x_\text{m}>0</math>. Suppose that for all <math>n</math>, the two random variables <math>\min\{X_1,\dotsc,X_n\}</math> and <math>(X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\}</math> are independent. Then the common distribution is a Pareto distribution.{{Citation needed|date=February 2012}}
===Geometric mean===
The geometric mean (G) is
几何平均(g)是
The [[geometric mean]] (''G'') is<ref name=Johnson1994>Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.</ref>
G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).
G = x _ text { m } exp left (frac {1}{ alpha } right).
: <math> G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).</math>
===Harmonic mean===
The harmonic mean (H) is of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions. Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto distribution generalizes Pareto Type IV.
调和平均数(h)是帕累托分布，称为帕累托 i 型、 II 型、 III 型、 IV 型和 Feller-帕累托分布。帕累托类型 IV 包含帕累托类型 i-III 作为特殊情况。Feller-帕累托分布推广了 Pareto 第四类。
The [[harmonic mean]] (''H'') is<ref name="Johnson1994"/>
<!--- In this context using x_m for the lower bound for the scale parameter is not meaningful, usual notation is \sigma --->
< ! ——在这种情况下，使用 x _ m 作为刻度参数的下限是没有意义的，通常的表示法是 sigma —— >
: <math> H = x_\text{m} \left( 1 + \frac{ 1 }{ \alpha } \right).</math>
The Pareto distribution hierarchy is summarized in the next table comparing the survival functions (complementary CDF).
下表总结了帕累托分布的层次结构，比较了生存函数(补充的 CDF)。
===Graphical representation===
The characteristic curved '[[long tail]]' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a [[log-log graph]], which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for ''x'' ≥ ''x''<sub>m</sub>,
When μ = 0, the Pareto distribution Type II is also known as the Lomax distribution.
当 μ = 0时，帕累托分布 II 型也称为洛马克斯分布。
:<math>\log f_X(x)= \log \left(\alpha\frac{x_\mathrm{m}^\alpha}{x^{\alpha+1}}\right) = \log (\alpha x_\mathrm{m}^\alpha) - (\alpha+1) \log x.</math>
In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.
在本节中，符号 xm (前面用来表示 x 的最小值)被 σ 替换。
Since ''α'' is positive, the gradient −(''α'' + 1) is negative.
{|class="wikitable" border="1"
{ | class = “ wikitable” border = “1”
|+Pareto distributions
| + 帕累托分布
==Related distributions==
! !! \overline{F}(x)=1-F(x) !! Support !! Parameters
!!!1-F (x) ! ！支持！参数
===Generalized Pareto distributions===
|-
|-
{{See also|Generalized Pareto distribution}}
| Type I
| 第一类
|| \left[\frac x \sigma \right]^{-\alpha}
[ frac x sigma right ] ^ {-alpha }
There is a hierarchy <ref name=arnold/><ref name=jkb94>Johnson, Kotz, and Balakrishnan (1994), (20.4).</ref> of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.<ref name=arnold/><ref name=jkb94/><ref name=kk03>{{cite book |author1=Christian Kleiber |author2=Samuel Kotz |lastauthoramp=yes |year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences |publisher=[[John Wiley & Sons|Wiley]] |isbn=978-0-471-15064-0|ref=harv| url=https://books.google.com/books?id=7wLGjyB128IC}}</ref> Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto<ref name=jkb94/><ref name=feller>{{cite book|last=Feller |first= W.| year=1971| title=An Introduction to Probability Theory and its Applications| volume=II| edition=2nd | location= New York|publisher=Wiley|page=50}} "The densities (4.3) are sometimes called after the economist ''Pareto''. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ ''Ax''<sup>−''α''</sup> as ''x'' → ∞."</ref> distribution generalizes Pareto Type IV.
|| x \ge \sigma
我们会找到他的
<!--- In this context using x_m for the lower bound for the scale parameter is not meaningful, usual notation is \sigma --->
|| \sigma > 0, \alpha
| | sigma > 0，alpha
|-
|-
====Pareto types I–IV====
| Type II
| 第二类
The Pareto distribution hierarchy is summarized in the next table comparing the [[survival function]]s (complementary CDF).
|| \left[1 + \frac{x-\mu} \sigma \right]^{-\alpha}
| | 左[1 + frac { x-mu } sigma right ] ^ {-alpha }
|| x \ge \mu
我们会找到他的
When ''μ'' = 0, the Pareto distribution Type II is also known as the [[Lomax distribution]].<ref>{{cite journal | last1 = Lomax | first1 = K. S. | year = 1954 | title = Business failures. Another example of the analysis of failure data | url = | journal = Journal of the American Statistical Association | volume = 49 | issue = 268| pages = 847–52 | doi=10.1080/01621459.1954.10501239}}</ref>
|| \mu \in \mathbb R, \sigma > 0, \alpha
数学天才，西格玛 > 0，阿尔法
|-
|-
In this section, the symbol ''x''<sub>m</sub>, used before to indicate the minimum value of ''x'', is replaced by ''σ''.
| Lomax
| 洛马克斯
|| \left[1 + \frac x \sigma \right]^{-\alpha}
[1 + frac x sigma right ] ^ {-alpha }
{|class="wikitable" border="1"
|| x \ge 0
我们会找到他的，我们会找到他
|+Pareto distributions
|| \sigma > 0, \alpha
| | sigma > 0，alpha
! !! <math> \overline{F}(x)=1-F(x)</math> !! Support !! Parameters
|-
|-
|-
| Type III
| 第三类
| Type I
|| \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1}
| | 左[1 + 左(frac { x-mu } sigma 右) ^ {1/gamma }右] ^ {-1}
|| <math>\left[\frac x \sigma \right]^{-\alpha}</math>
|| x \ge \mu
我们会找到他的
|| <math>x \ge \sigma</math>
|| \mu \in \mathbb R, \sigma, \gamma > 0
| | mu in mathbb r，sigma，gamma > 0
|| <math>\sigma > 0, \alpha</math>
|-
|-
|-
| Type IV
| 第 IV 类
| Type II
|| \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha}
| | 左[1 + 左(frac { x-mu } sigma 右) ^ {1/gamma }右] ^ {-alpha }
|| <math>\left[1 + \frac{x-\mu} \sigma \right]^{-\alpha}</math>
|| x \ge \mu
我们会找到他的
|| <math>x \ge \mu</math>
|| \mu \in \mathbb R, \sigma, \gamma > 0, \alpha
| | mu in mathbb r，sigma，gamma > 0，alpha
|| <math>\mu \in \mathbb R, \sigma > 0, \alpha</math>
|-
|-
|-
|-
|-
| Lomax
|}
|}
|| <math>\left[1 + \frac x \sigma \right]^{-\alpha}</math>
|| <math>x \ge 0</math>
The shape parameter α is the tail index, μ is location, σ is scale, γ is an inequality parameter. Some special cases of Pareto Type (IV) are
形状参数 α 是尾部指标，μ 是位置，σ 是标度，γ 是不等式参数。帕累托型(IV)的一些特殊情况是
|| <math>\sigma > 0, \alpha</math>
|-
P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),
P (IV)(sigma，sigma，1，alpha) = p (i)(sigma，alpha) ,
| Type III
P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),
P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1} </math>
P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).
p (IV)(mu，sigma，gamma，1) = p (III)(mu，sigma，gamma).
|| <math>x \ge \mu</math>
|| <math> \mu \in \mathbb R, \sigma, \gamma > 0</math>
The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments are finite for some δ > 0, as shown in the table below, where δ is not necessarily an integer.
均值的有限性、方差的存在性和有限性取决于尾指数 α (不等式指数 γ)。特别是，对于某些 δ > 0，分数阶 δ- 矩是有限的，如下表所示，其中 δ 不一定是整数。
|-
| Type IV
{|class="wikitable" border="1"
{ | class = “ wikitable” border = “1”
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha}</math>
|+Moments of Pareto I–IV distributions (case μ = 0)
帕累托 i-IV 分布的 | + 矩(μ = 0)
|| <math>x \ge \mu</math>
! !! \operatorname{E}[X] !! Condition !! \operatorname{E}[X^\delta] !! Condition
!!!操作员名称{ e }[ x ] ! ！条件！操作员名称{ e }[ x ^ delta ] ！环境影响评估条件
|| <math>\mu \in \mathbb R, \sigma, \gamma > 0, \alpha</math>
|-
|-
|-
| Type I
| 第一类
|-
|| \frac{\sigma \alpha}{\alpha-1}
| | frac { sigma alpha }{ alpha-1}
|}
|| \alpha > 1
| | alpha > 1
|| \frac{\sigma^\delta \alpha}{\alpha-\delta}
| | frac { sigma ^ delta alpha }{ alpha-delta }
The shape parameter ''α'' is the [[tail index]], ''μ'' is location, ''σ'' is scale, ''γ'' is an inequality parameter. Some special cases of Pareto Type (IV) are
|| \delta < \alpha
| | delta < alpha
|-
|-
::<math> P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),</math>
| Type II
| 第二类
::<math> P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),</math>
|| \frac{ \sigma }{\alpha-1}
| | frac { sigma }{ alpha-1}
::<math> P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).</math>
|| \alpha > 1
| | alpha > 1
|| \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}
| | frac { sigma ^ delta Gamma (alpha-delta) Gamma (1 + delta)}{ Gamma (alpha)}
The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index ''α'' (inequality index ''γ''). In particular, fractional ''δ''-moments are finite for some ''δ'' > 0, as shown in the table below, where ''δ'' is not necessarily an integer.
|| 0 < \delta < \alpha
| | 0 < delta < alpha
|-
|-
{|class="wikitable" border="1"
| Type III
| 第三类
|+Moments of Pareto I–IV distributions (case ''μ'' = 0)
|| \sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)
| | sigma Gamma (1-Gamma) Gamma (1 + Gamma)
! !! <math>\operatorname{E}[X]</math> !! Condition !! <math>\operatorname{E}[X^\delta]</math> !! Condition
|| -1<\gamma<1
| |-1 < γ < 1
|-
|| \sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)
| | sigma ^ delta Gamma (1-Gamma delta) Gamma (1 + Gamma delta)
| Type I
|| -\gamma^{-1}<\delta<\gamma^{-1}
| |-gamma ^ {-1} < delta < gamma ^ {-1}
|| <math>\frac{\sigma \alpha}{\alpha-1}</math>
|-
|-
|| <math>\alpha > 1</math>
| Type IV
| 第 IV 类
|| <math>\frac{\sigma^\delta \alpha}{\alpha-\delta}</math>
|| \frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)}
| | frac { sigma Gamma (alpha-Gamma) Gamma (1 + Gamma)}{ Gamma (alpha)}
|| <math> \delta < \alpha</math>
|| -1<\gamma<\alpha
| |-1 < gamma < alpha
|-
|| \frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}
| | frac { sigma ^ delta Gamma (alpha-Gamma delta) Gamma (1 + Gamma delta)}{ Gamma (alpha)}
| Type II
|| -\gamma^{-1}<\delta<\alpha/\gamma
| |-gamma ^ {-1} < delta < alpha/gamma
|| <math> \frac{ \sigma }{\alpha-1}</math>
|-
|-
|| <math>\alpha > 1</math>
|-
|-
|| <math> \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}</math>
|}
|}
|| <math>0 < \delta < \alpha</math>
|-
| Type III
Feller
费勒
|| <math>\sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)</math>
|| <math> -1<\gamma<1</math>
W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma
W = mu + sigma left (frac { u _ 1}{ u _ 2} right) ^ gamma
|| <math>\sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)</math>
|| <math>-\gamma^{-1}<\delta<\gamma^{-1}</math>
and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are
写出 w ~ FP (μ，σ，γ，δ1，δ2)。帕累托分布的特殊情况如下
|-
| Type IV
FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)
FP (sigma，sigma，1,1，alpha) = p (i)(sigma，alpha)
|| <math>\frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)}</math>
FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)
FP (mu，sigma，1,1，alpha) = p (II)(mu，sigma，alpha)
|| <math> -1<\gamma<\alpha</math>
FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)
FP (mu，sigma，gamma，1,1) = p (III)(mu，sigma，gamma)
|| <math>\frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}</math>
FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).
FP (mu，sigma，gamma，1，alpha) = p (IV)(mu，sigma，gamma，alpha).
|| <math>-\gamma^{-1}<\delta<\alpha/\gamma </math>
|-
|-
The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum xm and index α, then
美国帕累托分布协会与美国指数分布协会有如下关系。如果 x 是以最小 xm 和指数 α 为参数的 pareto 分布，则
|}
Y = \log\left(\frac{X}{x_\mathrm{m}}\right)
Y = log left (frac { x }{ x _ mathrm { m } right)
====Feller–Pareto distribution====
Feller<ref name=jkb94/><ref name=feller/> defines a Pareto variable by transformation ''U'' = ''Y''<sup>−1</sup> − 1 of a [[beta distribution|beta random variable]] ''Y'', whose probability density function is
is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then
指数分布的速率参数 α。等价，如果 y 是指数分布的速率 α，则
:<math> f(y) = \frac{y^{\gamma_1-1} (1-y)^{\gamma_2-1}}{B(\gamma_1, \gamma_2)}, \qquad 0<y<1; \gamma_1,\gamma_2>0,</math>
x_\mathrm{m} e^Y
2. x mathrm { m } e ^ y
where ''B''( ) is the [[beta function]]. If
is Pareto-distributed with minimum xm and index α.
是以最小 xm 和指数 α 为参数的 pareto 分布。
:<math> W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma>0, \gamma>0,</math>
This can be shown using the standard change-of-variable techniques:
这可以使用标准的变量更改技术来显示:
then ''W'' has a Feller–Pareto distribution FP(''μ'', ''σ'', ''γ'', ''γ''<sub>1</sub>, ''γ''<sub>2</sub>).<ref name=arnold/>
<math>
《数学》
\begin{align}
开始{ align }
If <math>U_1 \sim \Gamma(\delta_1, 1)</math> and <math>U_2 \sim \Gamma(\delta_2, 1)</math> are independent [[Gamma distribution|Gamma variables]], another construction of a Feller–Pareto (FP) variable is<ref>{{cite book |last=Chotikapanich |first=Duangkamon |title=Modeling Income Distributions and Lorenz Curves |chapter=Chapter 7: Pareto and Generalized Pareto Distributions |date=16 September 2008 |pages=121–22 |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}</ref>
\Pr(Y<y) & = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) \\
Pr (y < y) & = Pr left (log left (frac { x }{ x _ mathrm { m } right) < y right)
& = \Pr(X<x_\mathrm{m} e^y) = 1-\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1-e^{-\alpha y}.
1-left (frac { x _ mathrm { m } e ^ y }{ x _ mathrm { m } e ^ y } right) ^ alpha = 1-e ^ {-alpha y }.
:<math>W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma</math>
\end{align}
结束{ align }
</math>
数学
and we write ''W'' ~ FP(''μ'', ''σ'', ''γ'', ''δ''<sub>1</sub>, ''δ''<sub>2</sub>). Special cases of the Feller–Pareto distribution are
The last expression is the cumulative distribution function of an exponential distribution with rate α.
最后一个表达式是速率为 α 的累积分布函数的指数分布。
:<math>FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)</math>
:<math>FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)</math>
:<math>FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)</math>
The Pareto distribution and log-normal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the exponential distribution and normal distribution. (See the previous section.)
帕累托分布和对数正态分布是描述相同类型数量的替代分布。这两者之间的联系之一是，它们都是指数分布的随机变量，分别按照其他常见的分布，即指数分布和正态分布。(见上一节)
:<math>FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).</math>
===Relation to the exponential distribution===
The Pareto distribution is a special case of the generalized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the Lomax distribution as a special case. This family also contains both the unshifted and shifted exponential distributions.
帕累托分布分布是广义帕累托分布分布的一个特例，它是一族形式相似的分布，但包含一个额外的参数，使得分布的支撑要么在下面有界(在一个可变点) ，要么在上面和下面都有界(两者都是可变的) ，洛马克斯分布是一个特例。该族同时包含无移位和移位指数分布。
The Pareto distribution is related to the [[exponential distribution]] as follows. If ''X'' is Pareto-distributed with minimum ''x''<sub>m</sub> and index ''α'', then
The Pareto distribution with scale x_m and shape \alpha is equivalent to the generalized Pareto distribution with location \mu=x_m, scale \sigma=x_m/\alpha and shape \xi=1/\alpha. Vice versa one can get the Pareto distribution from the GPD by x_m = \sigma/\xi and \alpha=1/\xi.
具有尺度 x _ m 和形状 α 的帕累托分布相当于位置 mu = x _ m，尺度 σ = x _ m/α，形状 xi = 1/α 的广义帕累托分布。反之亦然，人们可以通过 x _ m = sigma/xi 和 α = 1/xi 从 GPD 得到帕累托分布。
: <math> Y = \log\left(\frac{X}{x_\mathrm{m}}\right) </math>
is [[exponential distribution|exponentially distributed]] with rate parameter ''α''. Equivalently, if ''Y'' is exponentially distributed with rate ''α'', then
{{Probability distribution
{概率分布
: <math> x_\mathrm{m} e^Y</math>
| name =Bounded Pareto
| name = Bounded Pareto
| type =density
类型 = 密度
is Pareto-distributed with minimum ''x''<sub>m</sub> and index ''α''.
| pdf_image =
图片来源: pdf
| cdf_image =
图片 | cdf/image =
This can be shown using the standard change-of-variable techniques:
| parameters =
参数 =
L > 0 location (real)<br />
L > 0 location (real) < br/>
: <math>
H > L location (real)<br />
H > l 位置(real) < br/>
\begin{align}
\alpha > 0 shape (real)
α > 0形状(实数)
\Pr(Y<y) & = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) \\
| support =L \leqslant x \leqslant H
| support = l leqslanx leqslanh
& = \Pr(X<x_\mathrm{m} e^y) = 1-\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1-e^{-\alpha y}.
| pdf =\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}
1-left (frac { l }{ h } right) ^ alpha }
\end{align}
| cdf =\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}
| cdf = frac {1-L ^ alpha x ^ {-alpha }{1-left (frac { l }{ h } right) ^ alpha }
</math>
| mean =
意味着
\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1 <br />
左(frac { l ^ alpha }{1-left (frac { l }{ h } right) ^ alpha } cdot left (frac { alpha }{ alpha-1} right) cdot left (frac {1}{ l ^ { alpha-1}-frac {1}{ h ^ { alpha-1}} right) ，alpha neq 1 < br/>
The last expression is the cumulative distribution function of an exponential distribution with rate ''α''.
\frac\ln\frac{H}{L}, \alpha=1
1.1.1
| median = L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}
| median = l left (1-frac {1}{2} left (1-left (1-left (frac { l }{ h } right) ^ alpha right)) ^ {-frac {1}{ alpha }}
===Relation to the log-normal distribution===
| mode =
2012年10月22日
The Pareto distribution and [[log-normal distribution]] are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the [[exponential distribution]] and [[normal distribution]]. (See [[#Relation_to_the_exponential_distribution|the previous section]].)
| variance =
| 方差 =
\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2
左(frac { l ^ alpha }{1-left (frac { l }{ h } right) ^ alpha } cdot left (frac { alpha }{ alpha-2} right) cdot left (frac {1}{ l ^ { alpha-2}-frac {1}{ h ^ { alpha-2}} right) ，alpha neq 2
===Relation to the generalized Pareto distribution===
\frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2
2{ h } ^ 2{ l } ^ 2}{ h } ^ 2-{ l } ^ 2} ln frac { h }{ l } ，alpha = 2
The Pareto distribution is a special case of the [[generalized Pareto distribution]], which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the [[Lomax distribution]] as a special case. This family also contains both the unshifted and shifted [[exponential distribution]]s.
(this is the second raw moment, not the variance)
(这是第二个原始时刻，不是方差)
| skewness = \frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha * (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j
| skewness = frac { l ^ { alpha }{1-left (frac { l }{ h } right) ^ { alpha } cdot frac { alpha * (l ^ { k-alpha }-h ^ { k-alpha })}{(alpha-k)}} ，alpha neq j
The Pareto distribution with scale <math>x_m</math> and shape <math>\alpha</math> is equivalent to the generalized Pareto distribution with location <math>\mu=x_m</math>, scale <math>\sigma=x_m/\alpha</math> and shape <math>\xi=1/\alpha</math>. Vice versa one can get the Pareto distribution from the GPD by <math>x_m = \sigma/\xi</math> and <math>\alpha=1/\xi</math>.
(this is the kth raw moment, not the skewness)
(这是最原始的时刻，不是最阴暗的时刻)
| kurtosis =
峰度 =
===Bounded Pareto distribution===
| entropy =
| 熵 =
{{See also|Truncated distribution}}
| mgf =
2012年10月22日
{{Probability distribution
| char =
2012年10月12日
| name =Bounded Pareto
}}
}}
| type =density
| pdf_image =
The bounded (or truncated) Pareto distribution has three parameters: α, L and H. As in the standard Pareto distribution α determines the shape. L denotes the minimal value, and H denotes the maximal value.
有界的(或截断的)帕累托分布有3个参数: α，l 和 h。就像标准的帕累托分布一样，α 决定形状。L 表示最小值，h 表示最大值。
| cdf_image =
| parameters =
The probability density function is
概率密度函数
<math>L > 0</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>H > L</math> [[location parameter|location]] ([[real numbers|real]])<br />
\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha},
1-left (frac { l }{ h } right) ^ alpha } ,
<math>\alpha > 0</math> [[shape parameter|shape]] (real)
| support =<math>L \leqslant x \leqslant H</math>
where L ≤ x ≤ H, and α > 0.
where L ≤ x ≤ H, and α > 0.
| pdf =<math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
| cdf =<math>\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
| mean =
If U is uniformly distributed on (0, 1), then applying inverse-transform method
如果 u 在(0,1)上是均匀分布的，则应用逆变换方法
<math>\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1 </math><br />
<math>\frac{{H}{L}}{{H}-{L}}\ln\frac{H}{L}, \alpha=1</math>
U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}
{1-l ^ alpha x ^ {-alpha }{1-(frac { l }{ h }) ^ alpha }
| median =<math> L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}</math>
x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}
X = left (- frac { u h ^ alpha-u l ^ alpha-h ^ alpha }{ h ^ alpha ^ alpha } right) ^ {-frac {1}{ alpha }}
| mode =
| variance =
is a bounded Pareto-distributed.
是一个有界的帕累托分布。
<math>\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2</math>
<math>\frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2</math>
(this is the second raw moment, not the variance)
| skewness = <math>\frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha * (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j </math>
The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.
对称帕累托分布和零对称帕累托分布的目的是捕捉一些具有尖锐概率峰值和对称长概率尾的特殊统计分布。这两种分布是从帕累托分布中推导出来的。长概率尾通常意味着概率衰减缓慢。在很多情况下，帕累托分布的工作都很合适。但是，如果分布具有两条慢衰减尾的对称结构，帕累托不能做到这一点。然后应用对称帕累托或零对称帕累托分布。
(this is the kth raw moment, not the skewness)
| kurtosis =
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:
对称累积分布函数帕累托分布系统(CDF)定义如下:
| entropy =
| mgf =
| char =
}}
The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is
给定一个独立样本 x = (x1，x2，... ，xn) ，帕累托分布参数 α 和 xm 的似然函数为
The bounded (or truncated) Pareto distribution has three parameters: ''α'', ''L'' and ''H''. As in the standard Pareto distribution ''α'' determines the shape. ''L'' denotes the minimal value, and ''H'' denotes the maximal value.
L(\alpha, x_\mathrm{m}) = \prod_{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod_{i=1}^n \frac {1}{x_i^{\alpha+1}}.
L (alpha，x mathrm { m }) = prod { i = 1} ^ n alpha frac { x mathrm { m } ^ alpha } = alpha ^ n ^ x mathrm { m }{ n ^ { n alpha } prod { i = 1} ^ n frac {1}{ x i ^ { alpha + 1}}.
The [[probability density function]] is
Therefore, the logarithmic likelihood function is
因此，对数似然函数是
: <math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>,
\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum_{i=1} ^n \ln x_i.
Ell (alpha，x _ mathrm { m }) = n ln alpha + n alpha ln x _ mathrm { m }-(alpha + 1) sum { i = 1} ^ n ln x _ i.
where ''L'' ≤ ''x'' ≤ ''H'', and ''α'' > 0.
It can be seen that \ell(\alpha, x_\mathrm{m}) is monotonically increasing with xm, that is, the greater the value of xm, the greater the value of the likelihood function. Hence, since x ≥ xm, we conclude that
可以看出，ell (alpha，x _ mathrum { m })随 xm 单调递增，即 xm 值越大，似然函数的值越大。因此，由于 x ≥ xm，我们得出结论:
====Generating bounded Pareto random variables====
If ''U'' is [[uniform distribution (continuous)|uniformly distributed]] on (0, 1), then applying inverse-transform method <ref>http://www.cs.bgu.ac.il/~mps042/invtransnote.htm</ref>
\widehat x_\mathrm{m} = \min_i {x_i}.
广义的 x mathrm { m } = min i { x i }。
:<math>U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}</math>
To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:
为了找到 α 的估计量，我们计算相应的偏导数，并确定它在哪里为零:
:<math>x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}</math>
\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.
{ partial alpha } = frac { n }{ alpha } + n ln x _ mathrm { m }-sum _ { i = 1} ^ n ln x _ i = 0.
is a bounded Pareto-distributed.{{Citation needed|date=February 2011}}
{{Clear}}
Thus the maximum likelihood estimator for α is:
因此，α 的最大似然估计量是:
===Symmetric Pareto distribution===
\widehat \alpha = \frac{n}{\sum _i \ln (x_i/\widehat x_\mathrm{m}) }.
广义 alpha = frac { n }{ sum _ i ln (x _ i/widehat x _ mathrm { m })}。
The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.<ref name=":0">{{Cite journal|last=Huang|first=Xiao-dong|date=2004|title=A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic|journal=IEEE Transactions on Broadcasting|volume=50|issue=3|pages=323–334|doi=10.1109/TBC.2004.834013}}</ref>
The expected statistical error is:
预期的统计错误是:
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:<ref name=":0" />
\sigma = \frac {\widehat \alpha} {\sqrt n}.
1.1.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.
<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over 2b-X}) ^a & X<b \\
Malik (1970) gives the exact joint distribution of (\hat{x}_\mathrm{m},\hat\alpha). In particular, \hat{x}_\mathrm{m} and \hat\alpha are independent and \hat{x}_\mathrm{m} is Pareto with scale parameter xm and shape parameter nα, whereas \hat\alpha has an inverse-gamma distribution with shape and scale parameters n − 1 and nα, respectively.
Malik (1970)给出了(hat { x } _ mathrm { m } ，hat alpha)的精确联合分布。特别地，hat { x } _ mathrm { m }和 hat alpha 是独立的，hat { x } _ mathrm { m }是带有尺度参数 xm 和形状参数 nα 的 Pareto 分布，而 hat alpha 是带有形状参数 n-1和尺度参数 nα 的反 γ 分布。
1- \tfrac{1}{2}(\tfrac{b}{X})^a& X\geq b
\end{cases}</math>
The corresponding probability density function (PDF) is:<ref name=":0" />
Vilfredo Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income. This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population controls 80% of the wealth. However, the 80-20 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his Cours d'économie politique indicates that about 30% of the population had about 70% of the income. The probability density function (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, net worth may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:
维尔弗雷多 · 帕雷托最初用这种分配来描述个人之间的财富分配，因为它似乎很好地表明了任何社会的大部分财富被该社会中较小比例的人拥有的方式。他还用它来描述收入的分配。这种观点有时被简单地表述为帕雷托法则或“80-20法则” ，即20% 的人口控制着80% 的财富。然而，80-20法则对应的是一个特定的 α 值，事实上，帕雷托在他的《政治经济学》中关于英国所得税的数据表明，大约30% 的人口拥有大约70% 的收入。本文开头的概率密度函数图表显示，人均拥有少量财富的“概率”或人口比例相当高，然后随着财富的增加而稳步下降。(然而，帕累托分布对于低端人群的财富是不现实的。事实上，净资产甚至可能是负值。)这种分配不仅限于描述财富或收入，而且在许多情况下，在从”小”到”大”的分配中找到了平衡。下面的例子有时被看作是近似帕累托分布的:
<!-- THESE TWO SEEM TO BELONG UNDER Zipf's law RATHER THAN THE PARETO DISTRIBUTION
< ! ——这两个似乎属于 Zipf 定律，而不是帕累托分布
<math>p(x) = {ab^a \over 2(b+\left\vert x-b \right\vert)^{a+1}},X\in R</math>
This distribution has two parameters: a and b. It is symmetric by b. Then the mathematic expectation is b. When, it has variance as following:
<math>E((x-b)^2)=\int_{-\infty}^{\infty} (x-b)^2p(x)dx={2b^2 \over (a-2)(a-1) }
Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq, see also distribution fitting ]]
使用累积帕累托(洛马克斯)分布最大一天降雨量[ CumFreq，参见分布拟合]
</math>
The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:
<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over 2b-X}) ^a & X<0 \\
1- \tfrac{1}{2}(\tfrac{b}{b+X})^a& X\geq 0
\end{cases}</math>
The Pareto distribution is a continuous probability distribution. Zipf's law, also sometimes called the zeta distribution, is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if the x values (incomes) are binned into N ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining x_m so that \alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)} where H(N,\alpha-1) is the generalized harmonic number. This makes Zipf's probability density function derivable from Pareto's.
帕累托分布是一个连续的概率分布。Zipf 定律，有时也被称为 zeta 分布，是一个离散的分布，将数值分离成一个简单的排名。两者都是负指数的简单幂律，按比例调整后，它们的累积分布等于1。如果 x 值(收入)被绑定到 n 个等级中，那么每个帕累托分布中的人数遵循1/等级模式，Zipf 就可以从中得到。通过定义 x _ m 使得 αx _ mathrm { m } ^ alpha = frac {1}{ h (n，α1)}其中 h (n，α1)是广义调和数，从而将分布归一化。这使得 Zipf 的概率密度函数可以从 Pareto 衍生出来。
The corresponding PDF is:
f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)}
F (x) = frac { alpha x _ mathrm { m } ^ alpha }{ x ^ { alpha + 1}} = frac {1}{ x ^ s h (n，s)}
<math>p(x) = {ab^a \over 2(b+\left\vert x \right\vert)^{a+1}},X\in R</math>
where s = \alpha-1 and x is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has f(x) probability of ranking x.
其中 s = alpha-1和 x 是一个整数，表示从1到 n 的等级，其中 n 是最高的收入等级。因此，从一个人口(或语言、互联网或国家)中随机选择的一个人(或单词、网站链接或城市)具有 f (x)排名概率 x。
This distribution is symmetric by zero. Parameter is related to the decay rate of probability and represents peak magnitude of probability.<ref name=":0" />
The "80-20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is \alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown to be mathematically equivalent:
根据“80-20法则” ，20% 的人收入占总收入的80% ，20% 最富裕的20% 收入占总收入的80% ，以此类推，当帕累托指数为 alpha = log 45 = cfrac { log 10}5 log {10}4}大约1.161时，恰好适用。这个结果可以从下面给出的洛伦兹曲线公式推导出来。此外，经证明以下几点在数学上是等价的:
===[[Multivariate Pareto distribution]]===
The univariate Pareto distribution has been extended to a [[multivariate distribution|multivariate]] Pareto distribution.<ref>{{cite journal
|last1=Rootzén|first1=Holger |last2=Tajvidi|first2=Nader
1-\frac{1}{\alpha}=\frac{\ln(1-p^n)}{\ln(1-(1-p)^n)}
1-frac {1}{ alpha } = frac { ln (1-p ^ n)}{ ln (1-(1-p) ^ n)}
|title=Multivariate generalized Pareto distributions |journal=Bernoulli|volume=12|year=2006|number=5 |pages=917–30
|doi=10.3150/bj/1161614952
This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
这不仅适用于收入，也适用于财富，或任何其他可以由这种分配模式。
|ref=harv |citeseerx=10.1.1.145.2991}}</ref>
This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.
这排除了0 < α ≤1的帕累托分布，如上所述，这种分布具有无限的期望值，因此不能合理地模拟收入分配。
==Statistical Inference==
===Estimation of parameters===
Price's square root law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that \alpha=1. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.
普莱斯的平方根定律有时被作为帕累托分布的性质或类似的性质提供。然而，该定律只适用于 alpha = 1的情况。请注意，在这种情况下，财富的总量和期望值是没有定义的，这个规则只是渐近地适用于随机样本。上面提到的扩展的帕雷托法则是一个更普遍的规则。
The [[likelihood function]] for the Pareto distribution parameters ''α'' and ''x''<sub>m</sub>, given an independent [[sample (statistics)|sample]] ''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>''), is
: <math>L(\alpha, x_\mathrm{m}) = \prod_{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod_{i=1}^n \frac {1}{x_i^{\alpha+1}}.</math>
Lorenz curves for a number of Pareto distributions. The case α = ∞ corresponds to perfectly equal distribution (G = 0) and the α = 1 line corresponds to complete inequality (G = 1)
一类 Pareto 分布的 Lorenz 曲线。情形 α = ∞对应完全等分布(g = 0) ，α = 1线对应完全不等式(g = 1)
Therefore, the logarithmic likelihood function is
The Lorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve L(F) is written in terms of the PDF f or the CDF F as
洛伦兹曲线常被用来描述收入和财富的分配。对于任何分布，洛伦兹曲线 l (f)用 PDF 表示，或用 CDF 表示
: <math>\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum_{i=1} ^n \ln x_i.</math>
L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}
L (f) = frac { int _ { x _ mathrm { m } ^ { x (f)} xf (x) ，dx }{ int _ { x _ mathrm { m } ^ infty xf (x) ，dx } = frac { int _ 0 ^ f x (f’) ，dF’}{ int _ 0 ^ 1 x (f’) ，dF’}
It can be seen that <math>\ell(\alpha, x_\mathrm{m})</math> is monotonically increasing with ''x''<sub>m</sub>, that is, the greater the value of ''x''<sub>m</sub>, the greater the value of the likelihood function. Hence, since ''x'' ≥ ''x''<sub>m</sub>, we conclude that
where x(F) is the inverse of the CDF. For the Pareto distribution,
其中 x (f)是 CDF 的倒数。对于帕累托分布来说,
: <math>\widehat x_\mathrm{m} = \min_i {x_i}.</math>
x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}}
x (f) = frac { x _ mathrm { m }{(1-F) ^ { frac {1}{ alpha }}}}
To find the [[estimator]] for ''α'', we compute the corresponding partial derivative and determine where it is zero:
and the Lorenz curve is calculated to be
计算出洛伦兹曲线为
: <math>\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.</math>
L(F) = 1-(1-F)^{1-\frac{1}{\alpha}},
1-(1-F) ^ {1-frac {1}{ alpha } ,
Thus the [[maximum likelihood]] estimator for ''α'' is:
For 0<\alpha\le 1 the denominator is infinite, yielding L=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
对于0 < alpha le 1，分母是无穷大，产生 l = 0。右图显示了一些帕累托分布的洛伦兹曲线的例子。
: <math>\widehat \alpha = \frac{n}{\sum _i \ln (x_i/\widehat x_\mathrm{m}) }.</math>
According to Oxfam (2016) the richest 62 people have as much wealth as the poorest half of the world's population. We can estimate the Pareto index that would apply to this situation. Letting ε equal 62/(7\times 10^9) we have:
根据牛津饥荒救济委员会(Oxfam)(2016)的数据，最富有的62个人拥有的财富相当于世界上最贫穷的一半人口的财富。我们可以估计适用于这种情况的帕累托指数。让 ε 等于62/(7乘以10 ^ 9)我们有:
L(1/2)=1-L(1-\varepsilon)
L (1/2) = 1-L (1-varepsilon)
The expected statistical error is:<ref>{{cite journal |author=M. E. J. Newman |year=2005 |title=Power laws, Pareto distributions and Zipf's law |journal=[[Contemporary Physics]] |volume=46 |issue=5 |pages=323–51| arxiv=cond-mat/0412004 |doi=10.1080/00107510500052444 |bibcode=2005ConPh..46..323N|s2cid=202719165 }}</ref>
or
或
1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}}
1-(1/2) ^ {1-frac {1}{ alpha }}} = varepsilon ^ {1-frac {1}{ alpha }}}
: <math>\sigma = \frac {\widehat \alpha} {\sqrt n}. </math>
<!--:\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(1-\frac{1}{\alpha})\ln\varepsilon
< ! -- : ln (1-(1/2) ^ {1-frac {1}{ alpha }}) = (1-frac {1}{ alpha }) ln varepsilon
\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(\ln\varepsilon/\ln 2)(1-\frac{1}{\alpha})\ln 2
Ln (1-(1/2) ^ {1-frac {1}{ alpha }}) = (ln varepsilon/ln 2)(1-frac {1}{ alpha }) ln 2
Malik (1970)<ref>{{cite journal |author=H. J. Malik |year=1970 |title=Estimation of the Parameters of the Pareto Distribution |journal=Metrika |volume=15|pages=126–132 |doi=10.1007/BF02613565 |s2cid=124007966 }}</ref> gives the exact joint distribution of <math>(\hat{x}_\mathrm{m},\hat\alpha)</math>. In particular, <math>\hat{x}_\mathrm{m}</math> and <math>\hat\alpha</math> are [[Independence (probability theory)|independent]] and <math>\hat{x}_\mathrm{m}</math> is Pareto with scale parameter ''x''<sub>m</sub> and shape parameter ''nα'', whereas <math>\hat\alpha</math> has an [[inverse-gamma distribution]] with shape and scale parameters ''n'' − 1 and ''nα'', respectively.
\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})
Ln (1-(1/2) ^ {1-frac {1}{ alpha }}) = -- lrb- ln varepsilon/ln 2) ln ((1/2) ^ {1-frac {1}{ alpha })
\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx(\ln\varepsilon/\ln 2)(1-(1/2)^{1-\frac{1}{\alpha}})
Ln (1-(1/2) ^ {1-frac {1}{ alpha }}) approx (ln varepsilon/ln 2)(1-(1/2) ^ {1-frac {1}{ alpha })
==Occurrence and applications==
-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\exp(-\ln(1-(1/2)^{1-\frac{1}{\alpha}}))\approx -\ln\varepsilon/\ln 2
- ln (1-(1/2) ^ {1-frac {1}{ alpha }}) exp (- ln (1-(1/2) ^ {1-frac {1}{ alpha }})) approx-ln varepsilon/ln 2
===General===
-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)
- ln (1-(1/2) ^ {1-frac {1}{ alpha }}) approx w (- ln varepsilon/ln 2)
[[Vilfredo Pareto]] originally used this distribution to describe the [[Distribution of wealth|allocation of wealth]] among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.<ref>Pareto, Vilfredo, ''Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino'', Librairie Droz, Geneva, 1964, pp. 299–345.</ref> This idea is sometimes expressed more simply as the [[Pareto principle]] or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.<ref>For a two-quantile population, where approximately 18% of the population owns 82% of the wealth, the [[Theil index]] takes the value 1.</ref> However, the 80-20 rule corresponds to a particular value of ''α'', and in fact, Pareto's data on British income taxes in his ''Cours d'économie politique'' indicates that about 30% of the population had about 70% of the income.{{citation needed|date=May 2019}} The [[probability density function]] (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, [[net worth]] may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:
where W is the Lambert W function. So
在那里 w 是朗伯W函数。所以
<!-- THESE TWO SEEM TO BELONG UNDER [[Zipf's law]] RATHER THAN THE PARETO DISTRIBUTION
(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))
(1/2) ^ {1-frac {1}{ alpha }} approx 1-exp (- w (- ln varepsilon/ln 2))
* Frequencies of words in longer texts (a few words are used often, lots of words are used infrequently)
{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2
{1-frac {1}{ alpha }}} approx-ln (1-exp (- w (- ln varepsilon/ln 2)))/ln 2
* Frequencies of [[Given name#Popularity distribution of given names|given names]] -->
\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)
(1-exp (- w (- ln varepsilon/ln 2))/ln 2)
* The sizes of human settlements (few cities, many hamlets/villages)<ref name="Reed">{{cite journal |citeseerx=10.1.1.70.4555 |first=William J. |last=Reed |title=The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions |journal=Communications in Statistics – Theory and Methods |volume=33 |issue=8 |pages=1733–53 |year=2004 |doi=10.1081/sta-120037438|s2cid=13906086 |display-authors=etal}}</ref><ref name="Reed2002">{{cite journal |first=William J. |last=Reed |title=On the rank‐size distribution for human settlements |journal=Journal of Regional Science |volume=42 |issue=1 |pages=1–17 |year=2002 |doi=10.1111/1467-9787.00247|s2cid=154285730 }}</ref>
-->The solution is that α equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.
解决方案是，α 大约等于1.15，这两个群体各拥有大约9% 的财富。但实际上，世界上最贫穷的69% 的成年人只拥有大约3% 的财富。
* File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)<ref name ="Reed" />
* [[Hard disk drive]] error rates<ref>{{cite journal |title=Understanding latent sector error and how to protect against them |url=http://www.usenix.org/event/fast10/tech/full_papers/schroeder.pdf |first1=Bianca |last1=Schroeder |first2=Sotirios |last2=Damouras |first3=Phillipa |last3=Gill |journal=8th Usenix Conference on File and Storage Technologies (FAST 2010)| date=2010-02-24 |accessdate=2010-09-10 |quote=We experimented with 5 different distributions (Geometric,Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ<sup>2</sup> statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.}}</ref>
The Gini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (α = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for \alpha\ge 1) to be
基尼系数是衡量洛伦兹曲线与等分布线之间的偏差，等分布线是一条连接[0,0]和[1,1]的线，右边的洛伦兹曲线用黑色(α = ∞)表示。具体来说，基尼系数是洛伦兹曲线和等分布线之间面积的两倍。然后计算出基尼系数帕累托分布的平均值(对于 alpha ge 1)为
* Clusters of [[Bose–Einstein condensate]] near [[absolute zero]]<ref name="Simon">{{cite journal|first2=Herbert A.|last2=Simon|author=Yuji Ijiri |title=Some Distributions Associated with Bose–Einstein Statistics|journal=Proc. Natl. Acad. Sci. USA|date=May 1975|volume=72|issue=5|pages=1654–57|pmc=432601|pmid=16578724|doi=10.1073/pnas.72.5.1654|bibcode=1975PNAS...72.1654I}}</ref>
[[File:FitParetoDistr.tif|thumb|250px|Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] ]]
G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1}
1-2 left (int _ 0 ^ 1L (f) ，dF right) = frac {1}{2 alpha-1}
* The values of [[oil reserves]] in oil fields (a few [[Giant oil and gas fields|large fields]], many [[Stripper well|small fields]])<ref name ="Reed" />
* The length distribution in jobs assigned to supercomputers (a few large ones, many small ones)<ref>{{Cite journal|last1=Harchol-Balter|first1=Mor|author1-link=Mor Harchol-Balter|last2=Downey|first2=Allen|date=August 1997|title=Exploiting Process Lifetime Distributions for Dynamic Load Balancing|url=https://users.soe.ucsc.edu/~scott/courses/Fall11/221/Papers/Sync/harcholbalter-tocs97.pdf|journal=ACM Transactions on Computer Systems|volume=15|issue=3|pages=253–258|doi=10.1145/263326.263344|s2cid=52861447}}</ref>
(see Aaberge 2005).
(见 Aaberge 2005)。
* The standardized price returns on individual stocks <ref name="Reed" />
* Sizes of sand particles <ref name ="Reed" />
* The size of meteorites
* Male dating success on Tinder [80% of females compete for the 20% most attractive males] <ref name="Medium.com">[https://medium.com/p/2ddf370a6e9a]</ref>
Random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0, 1], the variate T given by
利用逆变换采样生成随机样本。给定从单位区间上的均匀分布得到的随机变量 u (0,1] ，给出了 t 的变量 t
* Severity of large [[casualty (person)|casualty]] losses for certain lines of business such as general liability, commercial auto, and workers compensation.<ref>Kleiber and Kotz (2003): p. 94.</ref><ref>{{cite journal |last1=Seal |first1=H. |year=1980 |title=Survival probabilities based on Pareto claim distributions |journal=ASTIN Bulletin |volume=11 |pages=61–71|doi=10.1017/S0515036100006620 |doi-access=free }}</ref>
* Amount of time a user on [[Steam (software)|Steam]] will spend playing different games. (Some games get played a lot, but most get played almost never.) [https://docs.google.com/spreadsheets/d/1BDv2W4IsgxiAhhUznTbMSfRtcLia320Zq1HxzwhKao0/edit#gid=0]
T=\frac{x_\mathrm{m}}{U^{1/\alpha}}
1/alpha }}
* In [[hydrology]] the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.<ref>CumFreq, software for cumulative frequency analysis and probability distribution fitting [https://www.waterlog.info/cumfreq.htm]</ref> The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].
is Pareto-distributed. If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).
是帕累托分布的。如果 u 在[0,1]上是均匀分布的，则它可以与(1-u)交换。
===Relation to Zipf's law===
The Pareto distribution is a continuous probability distribution. [[Zipf's law]], also sometimes called the [[zeta distribution]], is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if the <math>x</math> values (incomes) are binned into <math>N</math> ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining <math>x_m</math> so that <math>\alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)}</math> where <math>H(N,\alpha-1)</math> is the [[Harmonic number#Generalized harmonic numbers|generalized harmonic number]]. This makes Zipf's probability density function derivable from Pareto's.
: <math>f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)}</math>
where <math>s = \alpha-1</math> and <math>x</math> is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has <math>f(x)</math> probability of ranking <math>x</math>.
===Relation to the "Pareto principle"===
The "[[80-20 law]]", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is <math>\alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161</math>. This result can be derived from the [[Lorenz curve]] formula given below. Moreover, the following have been shown<ref>{{cite journal |last1=Hardy |first1=Michael |year=2010 |title=Pareto's Law |journal=[[Mathematical Intelligencer]] |volume=32 |issue=3 |pages=38–43 |doi=10.1007/s00283-010-9159-2|s2cid=121797873 }}</ref> to be mathematically equivalent:
* Income is distributed according to a Pareto distribution with index ''α'' > 1.
* There is some number 0 ≤ ''p'' ≤ 1/2 such that 100''p'' % of all people receive 100(1 − ''p'')% of all income, and similarly for every real (not necessarily integer) ''n'' > 0, 100''p<sup>n</sup>'' % of all people receive 100(1 − ''p'')<sup>''n''</sup> percentage of all income. ''α'' and ''p'' are related by
:: <math>1-\frac{1}{\alpha}=\frac{\ln(1-p^n)}{\ln(1-(1-p)^n)}</math>
This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
This excludes Pareto distributions in which 0 < ''α'' ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.
===Relation to Price's law===
[[Derek J. de Solla Price#Scientific contributions|Price's square root law]] is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that <math>\alpha=1</math>. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.
===Lorenz curve and Gini coefficient===
| title=Ecrits sur la courbe de la répartition de la richesse
| title=Ecrits sur la courbe de la répartition de la richesse
[[File:ParetoLorenzSVG.svg|thumb|325px|Lorenz curves for a number of Pareto distributions. The case ''α'' = ∞ corresponds to perfectly equal distribution (''G'' = 0) and the ''α'' = 1 line corresponds to complete inequality (''G'' = 1)]]
| first=Vilfredo
第一名: 维尔弗雷多
| last=Pareto
最后 = 帕累托
The [[Lorenz curve]] is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve ''L''(''F'') is written in terms of the PDF ''f'' or the CDF ''F'' as
| editor=Librairie Droz
编辑: Librairie Droz
| year=1965
1965年
:<math>L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}</math>
| pages=48
48
| series=Œuvres complètes : T. III
| series=Œuvres complètes : T. III
where ''x''(''F'') is the inverse of the CDF. For the Pareto distribution,
| isbn=9782600040211}}
9782600040211}
:<math>x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}}</math>
and the Lorenz curve is calculated to be
| first=Vilfredo
第一名: 维尔弗雷多
| last=Pareto
最后 = 帕累托
:<math>L(F) = 1-(1-F)^{1-\frac{1}{\alpha}},</math>
| year=1896
1896年
| title=Cours d'économie politique
| title=Cours d'économie politique
For <math>0<\alpha\le 1</math> the denominator is infinite, yielding ''L''=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
| doi=10.1177/000271629700900314| s2cid=143528002
10.1177/000271629700900314 | s2cid = 143528002
}}
}}
According to [[Oxfam]] (2016) the richest 62 people have as much wealth as the poorest half of the world's population.<ref>{{cite web|title=62 people own the same as half the world, reveals Oxfam Davos report|url=https://www.oxfam.org/en/pressroom/pressreleases/2016-01-18/62-people-own-same-half-world-reveals-oxfam-davos-report|publisher=Oxfam|date=Jan 2016}}</ref> We can estimate the Pareto index that would apply to this situation. Letting ε equal <math>62/(7\times 10^9)</math> we have:
:<math>L(1/2)=1-L(1-\varepsilon)</math>
or
:<math>1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}}</math>
<!--:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(1-\frac{1}{\alpha})\ln\varepsilon</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(\ln\varepsilon/\ln 2)(1-\frac{1}{\alpha})\ln 2</math>
| url=http://www3.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf
Http://www3.unisi.it/eventi/ginilorenz05/25%20may%20paper/paper_aaberge.pdf
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})</math>
| title=Gini's Nuclear Family
| title = Gini 的核心家庭
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx(\ln\varepsilon/\ln 2)(1-(1/2)^{1-\frac{1}{\alpha}})</math>
| first=Rolf
第一个 = Rolf
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\exp(-\ln(1-(1/2)^{1-\frac{1}{\alpha}}))\approx -\ln\varepsilon/\ln 2</math>
| last=Aabergé
| last=Aabergé
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)</math>
| conference=International Conference to Honor Two Eminent Social Scientists
两位杰出社会科学家荣誉国际会议
where ''W'' is the [[Lambert W function]]. So
| date=May 2005}}
2005年5月
:<math>(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))</math>
:<math>{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2</math>
:<math>\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)</math>
| url=https://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
Https://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
-->The solution is that ''α'' equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.<ref>{{cite web|title=Global Wealth Report 2013|url=https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|publisher=Credit Suisse|page=22|date=Oct 2013|access-date=2016-01-24|archive-url=https://web.archive.org/web/20150214155424/https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|archive-date=2015-02-14|url-status=dead}}</ref>
| title=Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes
| title = 万维网流量的自相似性: 证据和可能的原因
| first1=Mark E.
1 = Mark e.
The [[Gini coefficient]] is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (''α'' = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for <math>\alpha\ge 1</math>) to be
| last1=Crovella
1 = Crovella
| author-link1=Mark Crovella
1 = Mark Crovella
:<math>G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1}</math>
| first2=Azer
2 = Azer
| last2=Bestavros
2 = Bestavros
(see Aaberge 2005).
| conference=IEEE/ACM Transactions on Networking
会议 = IEEE/ACM网络学报
| volume=5
5
==Computational methods==
| number=6
6
===Random sample generation===
| pages=835–846
| 页数 = 835-846
Random samples can be generated using [[inverse transform sampling]]. Given a random variate ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] on the unit interval (0, 1], the variate ''T'' given by
| date=December 1997}}
| date = December 1997}
:<math>T=\frac{x_\mathrm{m}}{U^{1/\alpha}}</math>
is Pareto-distributed.<ref>{{cite book |last=Tanizaki |first=Hisashi |title=Computational Methods in Statistics and Econometrics |year=2004 |page=133 |publisher=CRC Press |url=https://books.google.com/books?id=pOGAUcn13fMC|isbn=9780824750886 }}</ref> If ''U'' is uniformly distributed on [0, 1), it can be exchanged with (1 − ''U'').
==See also==
* [[Bradford's law]]
* [[Gutenberg–Richter law]]
* [[Matthew effect]]
Category:Actuarial science
类别: 精算
* [[Pareto analysis]]
Category:Continuous distributions
类别: 连续分布
* [[Pareto efficiency]]
Category:Power laws
分类: 权力法则
* [[Pareto interpolation]]
Category:Probability distributions with non-finite variance
范畴: 非有限方差的概率分布
* [[Power law#Power-law probability distributions|Power law probability distributions]]
Category:Exponential family distributions
类别: 指数族分布
* [[Sturgeon's law]]
Category:Vilfredo Pareto
类别: Vilfredo Pareto
<noinclude>
<small>This page was moved from [[wikipedia:en:Pareto distribution]]. Its edit history can be viewed at [[Pareto分布/edithistory]]</small></noinclude>
[[Category:待整理页面]]