Pareto分布

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模板:Probability distribution


The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,[1] 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,[2] 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". 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]\displaystyle{ \alpha }[/math] (α = log45 ≈ 1.16). While [math]\displaystyle{ \alpha }[/math] is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena[3] and human activities.[4]

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". 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 维尔弗雷多·帕累托Vilfredo Pareto命名的帕累托概率分布,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律概率。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,或“80-20法则” ,有时也被称为 “马太原则”。<!——我们能找到比https://youtu.be/5WX9UEYZsR8 at 2'10"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定义

If X is a random variable with a Pareto (Type I) distribution,[5] 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 是一个具有 帕累托分布 (1 型)Pareto (Type i)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Tail function),由以下给出


[math]\displaystyle{ \overline{F}(x) = \Pr(X\gt x) = \begin{cases} \lt math\gt \overline{F}(x) = \Pr(X\gt x) = \begin{cases} \lt math \gt overline { f }(x) = Pr (x \gt 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 \lt x_\mathrm{m}, 1 & x \lt x_\mathrm{m}, 1 & x \lt x mathrm { m } , \end{cases} \end{cases} 结束{ cases } }[/math]

</math>



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.

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”的最小可能值(必然为正),“α”是一个正参数。帕累托 i 型分布 Pareto Type I distribution的特征值是比例参数xm形状参数“α”,即所谓的“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 累积分布函数

From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm 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Cumulative distribution function


[math]\displaystyle{ F_X(x) = \begin{cases} \lt math\gt F_X(x) = \begin{cases} \lt math \gt 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 \lt x_\mathrm{m}. 0 & x \lt x_\mathrm{m}. 0 & x \lt x mathrm { m }. \end{cases} }[/math]

\end{cases}</math>

结束{ cases } </math >

Probability density function概率密度函数

It follows (by 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: 概率密度函数为


[math]\displaystyle{ f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x \lt 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.

当在线性轴上绘制时,分布曲线为熟悉的J形曲线,该曲线渐近地接近每个正交轴。曲线的所有段都是自相似的(取决于适当的比例因子)。在双对数图中绘制时,分布用直线表示。

Properties性质

Moments and characteristic function矩与特征函数

[math]\displaystyle{ \operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\ \lt math\gt \operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\ (x) = begin { cases } infty & alpha le 1, \frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha\gt 1. \frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha\gt 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 >


[math]\displaystyle{ \operatorname{Var}(X)= \begin{cases} \lt math\gt \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\gt 2. \left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha\gt 2. 左(frac { x _ mathrm { m }{ alpha-1}右) ^ 2 frac { alpha-2} & alpha \gt 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.)


[math]\displaystyle{ \mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha\gt 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 }


[math]\displaystyle{ 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]\displaystyle{ 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。


[math]\displaystyle{ \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(ax) 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.

beta = mean * Sqr(mean ^ 2 + var) / (Sqr(var) + Sqr(mean ^ 2 + var)) -->

Beta = mean * Sqr (mean ^ 2 + var)/(Sqr (var) + Sqr (mean ^ 2 + var)) -- >

[6]


Conditional distributions条件分布

The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number 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Pareto分布随机变量的条件概率分布,如果它大于或等于某个特定的数[math]\displaystyle{ x_1 }[/math] > [math]\displaystyle{ x_\text{m} }[/math],则它是具有相同 Pareto指数 Pareto分布,但是具有最小的[math]\displaystyle{ x_1 }[/math]而不是[math]\displaystyle{ x_\text{m} }[/math]

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]\displaystyle{ x_1 }[/math] exceeding [math]\displaystyle{ x_\text{m} }[/math], is a Pareto distribution with the same Pareto index [math]\displaystyle{ \alpha }[/math] but with minimum [math]\displaystyle{ x_1 }[/math] instead of [math]\displaystyle{ x_\text{m} }[/math].

A characterization theorem一个特征定理

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[math]\displaystyle{ X_1, X_2, X_3, \dotsc }[/math]是独立同分布的随机变量,其概率分布在区间[math]\displaystyle{ [x_\text{m},\infty) }[/math] 上,对于某些[math]\displaystyle{ x_\text{m}\gt 0 }[/math]成立。假设对于所有[math]\displaystyle{ n }[/math],两个随机变量[math]\displaystyle{ \min\{X_1,\dotsc,X_n\} }[/math] and [math]\displaystyle{ (X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\} }[/math] 相互独立,那么其公共分布就是 帕累托分布[citation needed]

Suppose [math]\displaystyle{ X_1, X_2, X_3, \dotsc }[/math] are independent identically distributed random variables whose probability distribution is supported on the interval [math]\displaystyle{ [x_\text{m},\infty) }[/math] for some [math]\displaystyle{ x_\text{m}\gt 0 }[/math]. Suppose that for all [math]\displaystyle{ n }[/math], the two random variables [math]\displaystyle{ \min\{X_1,\dotsc,X_n\} }[/math] and [math]\displaystyle{ (X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\} }[/math] are independent. Then the common distribution is a Pareto distribution.[citation needed]

Geometric mean几何平均数

The geometric mean (G) is

几何平均数(G)

The geometric mean (G) is[7] 几何平均值 (“G”)是[7]


 G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).

G = x _ text { m } exp left (frac {1}{ alpha } right).

[math]\displaystyle{ G = x_\text{m} \exp \left( \frac{1}{\alpha} \right). }[/math]


Harmonic mean调和平均数

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.

调何平均数(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 第四型。

The harmonic mean (H) is[7]


< ! ——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]\displaystyle{ 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图示法

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 xxm,

当在线性标度上绘制时,“长尾”分布特征曲线在对数曲线图上绘制时,掩盖了函数潜在的简单性,然后采用负梯度的直线形式:根据概率密度函数的公式,对于xxm

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 Pareto distribution Type II也称为洛马克斯分布Lomax distribution


[math]\displaystyle{ \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.

由于“α”为正,因此梯度−(α + 1)为负。

{ | class = “ wikitable” border = “1”

模板:另见广义帕累托分布

In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ. 在本节中,之前用于表示“x”最小值的符号“x”m将替换为“σ”。
Pareto distributions
+ 帕累托分布

Related distributions相关分布

\overline{F}(x)=1-F(x) Support Parameters 1-F (x) ! !支持!参数

Generalized Pareto distributions广义帕累托分布

Type I 第一类


\left[\frac x \sigma \right]^{-\alpha}

[ frac x sigma right ] ^ {-alpha }

There is a hierarchy [5][8] of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.[5][8][9] Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto[8][10] distribution generalizes Pareto Type IV. 帕累托分布有一个层次结构[5][8]帕累托分布被称为帕累托类型I、II、III、IV,费勒-帕累托分布。[5][8][9]Pareto类型IV包含Pareto类型I-III作为特殊情况。费勒-帕累托[8]引用错误:没有找到与</ref>对应的<ref>标签

当“μ”=0时,帕累托分布类型II也称为Lomax distribution[11]

\mu \in \mathbb R, \sigma > 0, \alpha



Lomax 洛马克斯


\left[1 + \frac x \sigma \right]^{-\alpha}

[1 + frac x sigma right ] ^ {-alpha }

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]\displaystyle{ \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]\displaystyle{ \left[\frac x \sigma \right]^{-\alpha} }[/math] x \ge \mu

我们会找到他的

[math]\displaystyle{ x \ge \sigma }[/math] \mu \in \mathbb R, \sigma, \gamma > 0 mu in mathbb r,sigma,gamma > 0 [math]\displaystyle{ \sigma \gt 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]\displaystyle{ \left[1 + \frac{x-\mu} \sigma \right]^{-\alpha} }[/math] x \ge \mu

我们会找到他的

[math]\displaystyle{ x \ge \mu }[/math] \mu \in \mathbb R, \sigma, \gamma > 0, \alpha mu in mathbb r,sigma,gamma > 0,alpha [math]\displaystyle{ \mu \in \mathbb R, \sigma \gt 0, \alpha }[/math]
Lomax

|| [math]\displaystyle{ \left[1 + \frac x \sigma \right]^{-\alpha} }[/math]

|| [math]\displaystyle{ 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]\displaystyle{ \sigma \gt 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]\displaystyle{ \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]\displaystyle{ x \ge \mu }[/math]

|| [math]\displaystyle{ \mu \in \mathbb R, \sigma, \gamma \gt 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”
[math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \mu \in \mathbb R, \sigma, \gamma \gt 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 形状参数“α”是尾部索引,“μ”是位置,“σ”是尺度,“γ”是不等式参数。帕累托类型(IV)的一些特殊情况是 || \delta < \alpha

| | delta < alpha


|-

|-

[math]\displaystyle{ P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha), }[/math]

| Type II

| 第二类

[math]\displaystyle{ P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha), }[/math]

|| \frac{ \sigma }{\alpha-1}

| | frac { sigma }{ alpha-1}

[math]\displaystyle{ 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. 均值的有限性、方差的存在性和有限性取决于尾部指数α(不等式指数γ)。特别是,对于某些“δ”>0,分数“δ”矩是有限的,如下表所示,其中“δ”不一定是整数。

|| 0 < \delta < \alpha

| | 0 < delta < alpha


|-

|-

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]\displaystyle{ \operatorname{E}[X] }[/math] Condition [math]\displaystyle{ \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]\displaystyle{ \frac{\sigma \alpha}{\alpha-1} }[/math]
[math]\displaystyle{ \alpha \gt 1 }[/math] Type IV 第 IV 类 [math]\displaystyle{ \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]\displaystyle{ \delta \lt \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]\displaystyle{ \frac{ \sigma }{\alpha-1} }[/math]
[math]\displaystyle{ \alpha \gt 1 }[/math]
[math]\displaystyle{ \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)} }[/math]

|}

|| [math]\displaystyle{ 0 \lt \delta \lt \alpha }[/math]

|-

| Type III

Feller

费勒

|| [math]\displaystyle{ \sigma\Gamma(1-\gamma)\Gamma(1 + \gamma) }[/math]

|| [math]\displaystyle{ -1\lt \gamma\lt 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]\displaystyle{ \sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta) }[/math]

|| [math]\displaystyle{ -\gamma^{-1}\lt \delta\lt \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]\displaystyle{ \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]\displaystyle{ -1\lt \gamma\lt \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]\displaystyle{ \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]\displaystyle{ -\gamma^{-1}\lt \delta\lt \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费勒-帕累托分布

Feller[8][10] defines a Pareto variable by transformation U = Y−1 − 1 of a beta random variable Y, whose probability density function is

费勒Feller[8][12]通过β随机变量 Y的转换U = Y−1 − 1 定义一个Pareto变量,其概率密度函数为

is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then

与速率参数α呈指数分布。等价地,如果Y与速率α呈指数分布,则



[math]\displaystyle{ f(y) = \frac{y^{\gamma_1-1} (1-y)^{\gamma_2-1}}{B(\gamma_1, \gamma_2)}, \qquad 0\lt y\lt 1; \gamma_1,\gamma_2\gt 0, }[/math]
 x_\mathrm{m} e^Y

2. x mathrm { m } e ^ y


where B( ) is the beta function. If

其中 B( )是β函数。如果

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]\displaystyle{ W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma\gt 0, \gamma\gt 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, γ1, γ2).[5]

那么“W”具有Feller-Pareto分布FP(μ, σ, γ, γ1, γ2)。[5]

[math]\displaystyle{ 

《数学》



\begin{align}

开始{ align }

If \lt math\gt U_1 \sim \Gamma(\delta_1, 1) }[/math] and [math]\displaystyle{ U_2 \sim \Gamma(\delta_2, 1) }[/math] are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is[13]

如果[math]\displaystyle{ U_1 \sim \Gamma(\delta_1, 1) }[/math][math]\displaystyle{ U_2 \sim \Gamma(\delta_2, 1) }[/math]是相互独立的[[伽马分布|伽马变量],Feller-Pareto(FP)变量的另一个构造是[14]

\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]\displaystyle{ 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, δ1, δ2). Special cases of the Feller–Pareto distribution are

我们写为W ~ FP(μ, σ, γ, δ1, δ2)。Feller-Pareto分布的特殊情况是

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]\displaystyle{ FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha) }[/math]
[math]\displaystyle{ FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha) }[/math]
[math]\displaystyle{ 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同一类型变量的替代分布。二者之间的联系之一是,它们都是服从其他常见分布的随机变量的指数分布,分别服从指数分布和正态分布。(请参阅上一节。)

[math]\displaystyle{ FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha). }[/math]

Relation to the exponential distribution与指数分布的关系

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Pareto distribution 广义帕累托分布Generalized Pareto distribution 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有下界(在一个可变点处) ,要么上下都有界(两者都可变) , 洛马克斯分布Lomax distribution 是一个特例。该族同时包含无移位和移位指数分布Unshifted and Shifted exponential distributions

The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum xm and index α, then

帕累托分布与指数分布的关系如下。如果“X”是帕累托分布,具有最小xm 和指数 α,则

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.

具有规模 x_m 和形状 \alpha的帕累托分布相当于位置 mu = x _ m,规模 \sigma=x_m/\alpha 和 形状 \xi=1/\alpha 的广义帕累托分布。反之亦然,人们可以通过 x_m = \sigma/\xi 和 \alpha=1/\xi 从 GPD 得到帕累托分布。

[math]\displaystyle{ Y = \log\left(\frac{X}{x_\mathrm{m}}\right) }[/math]


is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then

指数分布,速率参数为“α”。等价地,如果“Y”与速率“α”呈指数分布,则

{{Probability distribution

{概率分布

[math]\displaystyle{ x_\mathrm{m} e^Y }[/math]
| name       =Bounded Pareto

| 名称=有界帕累托


| type       =density

类型 = 密度

is Pareto-distributed with minimum xm and index α. 帕累托分布的最小值是“x”m和索引“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: 这可以用变量技术的标准变化来表示:

| parameters =

参数 =


L > 0 location (real)

L > 0 location (real) < br/>

[math]\displaystyle{ H \gt L location (real)\lt br /\gt H \gt l 位置(real) \lt br/\gt \begin{align} \alpha \gt 0 shape (real) α \gt 0形状(实数) \Pr(Y\lt y) & = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)\lt y\right) \\ | support =L \leqslant x \leqslant H | 支持= L \莱克斯兰 x \莱克斯兰 H & = \Pr(X\lt 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       =

|中值 =


\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

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与对数正态分布的关系

| 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 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。两者之间的联系之一是,它们都是服从其他常见分布的随机变量的指数分布,分别服从指数分布正态分布。(见上一节。)

| 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与广义帕累托分布的关系

\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 distributions.

帕累托分布是广义帕累托分布的一个特例,它是一个形式相似的分布族,但包含一个额外的参数,使得分布的支持要么有下界(在可变点),要么在上下都有界(其中两者都是可变的),以洛马克斯分布为特例。该族还包含非移位和移位的指数分布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]\displaystyle{ x_m }[/math] and shape [math]\displaystyle{ \alpha }[/math] is equivalent to the generalized Pareto distribution with location [math]\displaystyle{ \mu=x_m }[/math], scale [math]\displaystyle{ \sigma=x_m/\alpha }[/math] and shape [math]\displaystyle{ \xi=1/\alpha }[/math]. Vice versa one can get the Pareto distribution from the GPD by [math]\displaystyle{ x_m = \sigma/\xi }[/math] and [math]\displaystyle{ \alpha=1/\xi }[/math].

具有规模[math]\displaystyle{ x_m }[/math]和形状[math]\displaystyle{ \alpha }[/math]的Pareto分布相当于具有位置[math]\displaystyle{ \mu=x_m }[/math],规模[math]\displaystyle{ \sigma=x_m/\alpha }[/math] 和形状 [math]\displaystyle{ \xi=1/\alpha }[/math]的广义Pareto分布。反之亦然,我们可以从GPD通过[math]\displaystyle{ x_m = \sigma/\xi }[/math][math]\displaystyle{ \alpha=1/\xi }[/math]得到Pareto分布。

(this is the kth raw moment, not the skewness)

(这是第k个原始时刻,不是偏斜)


| kurtosis   =

峰度 =

Bounded Pareto distribution有界帕累托分布

| entropy    =

| 熵 =

模板:另请参阅

| mgf        =

2012年10月22日

{{Probability distribution 模板:概率分布

}}

| type       =density

类型=密度

| 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 =

参数= The probability density function is

概率密度函数为

[math]\displaystyle{ L \gt 0 }[/math] location (real)

[math]\displaystyle{ H \gt L }[/math] location (real)

\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha},

1-left (frac { l }{ h } right) ^ alpha } ,

[math]\displaystyle{ \alpha \gt 0 }[/math] shape (real)

| support    =[math]\displaystyle{ L \leqslant x \leqslant H }[/math]

where L ≤ x ≤ H, and α > 0.

where L ≤ x ≤ H, and α > 0.

| pdf        =[math]\displaystyle{ \frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha} }[/math]
| cdf        =[math]\displaystyle{ \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]\displaystyle{ \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]

[math]\displaystyle{ \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]\displaystyle{  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]\displaystyle{ \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]\displaystyle{ \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) (这是第二个原始时刻,不是方差)

| skewness   = [math]\displaystyle{ \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Pareto分布 零对称Pareto分布的目的是捕捉具有尖峰和对称长概率尾的特殊统计分布。这两种分布是由帕累托分布导出的。长概率尾通常意味着概率衰减缓慢。帕累托分布在许多情况下能很好地拟合。但是,如果分布具有两个慢衰减尾的对称结构,则Pareto分布无法做到这点。于是此时应用对称帕累托分布或零对称帕累托分布。

(this is the kth raw moment, not the skewness) (这是第k个原始时刻,不是偏斜)

| 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Cumulative distribution function (CDF)分布系统定义如下:

| 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Likelihood function

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(或截断)Pareto分布有三个参数:“α”、“L”和“H”。在标准帕累托分布中,α决定了形状L”表示最小值,“H”表示最大值。

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

概率密度函数

Therefore, the logarithmic likelihood function is

因此, 对数似然函数 Logarithmic Likelihood function

[math]\displaystyle{ \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.

其中L ≤ x ≤ H, 且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生成有界Pareto随机变量

If U is uniformly distributed on (0, 1), then applying inverse-transform method [15]

如果“U”在(0,1)上为均匀分布,则应用反变换方法[16]

\widehat x_\mathrm{m} = \min_i {x_i}.

广义的x_\mathrm{m} = \min_i {x_i}。


[math]\displaystyle{ 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]\displaystyle{ 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] 是一个有界的帕累托分布。[citation needed]

Thus the maximum likelihood estimator for α is:

因此,α 的最大似然估计量是:

Symmetric Pareto distribution对称帕累托分布

\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.[17]

对称Pareto分布和零对称Pareto分布的目的是捕捉具有尖峰和对称长概率尾的特殊统计分布。这两种分布是由帕累托分布导出的。长概率尾通常意味着概率衰减缓慢。帕累托分布在许多情况下可以拟合。但是,如果分布具有两个慢衰减尾的对称结构,则Pareto无法做到这一点。于是此时应用对称帕累托或零对称帕累托分布

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:[17] 对称帕累托分布的累积分布函数(CDF)定义如下:[17]


\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]\displaystyle{ F(X) = P(x \lt X ) = \begin{cases} \tfrac{1}{2}({b \over 2b-X}) ^a & X\lt 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:[17]

相应的概率密度函数(PDF)为:[17]

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:

Vilfredo Pareto最初用这种分布来描述个人之间的财富分配,因为它似乎很好地表明,任何社会的大部分财富都是由社会中较小比例的人拥有的。他还用它来描述收入分配。这种观点有时更简单地表述为帕累托原则或“80-20法则”,即20%的人口控制着80%的财富。然而,80-20规则对应于一个特定的α值,事实上,帕累托在其经济政治课程中关于英国所得税的数据表明,大约30%的人口拥有约70%的收入。本文开头的概率密度函数(PDF)图显示,人均拥有少量财富的人口的“概率”或比例相当高,然后随着财富的增加而稳步下降。(然而,对于低端财富而言,帕累托分布并不现实。事实上,净资产甚至可能是负的。)这种分布不仅限于描述财富或收入,而且在许多情况下,在从“小”到“大”的分配中能够找到平衡。以下示例有时被视为近似帕累托分布:


\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)[18][19]
  • 人类居住区的规模(少数城市,许多村庄/村庄)[18][19]

-->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)[18]
  • 使用TCP协议的Internet流量的文件大小分布(许多较小的文件,少数较大的文件)[20]

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

文件:FitParetoDistr.tif
Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting

[[文件:FitParetoDistr.tif|thumb | 250px |拟合累积帕累托(Lomax)分布到最大一天降雨量,使用CumFreq,另见分布拟合]]

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 length distribution in jobs assigned to supercomputers (a few large ones, many small ones)[24]
  • 分配给超级计算机的作业长度分布(一些大型计算机,许多小型计算机)[25]

(see Aaberge 2005).

(见 Aaberge 2005)。

  • The standardized price returns on individual stocks [18]
  • 个股的标准化价格回报率[20]
  • Sizes of sand particles [18]
  • The size of meteorites
  • 陨石的大小
  • Male dating success on Tinder [80% of females compete for the 20% most attractive males] [26]
  • 在Tinder上,男性约会成功【80%的女性竞争20%最具吸引力的男性】[26]

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 losses for certain lines of business such as general liability, commercial auto, and workers compensation.[27][28]
  • 某些业务领域(如一般责任险、商用车等)的大额伤亡损失的严重程度,[29][30]
  • Amount of time a user on Steam will spend playing different games. (Some games get played a lot, but most get played almost never.) [1]
  • 用户在 Steam上玩不同游戏的时间。(有些游戏经常玩,但大多数几乎从不玩。)[2]

T=\frac{x_\mathrm{m}}{U^{1/\alpha}}

1/alpha }}

  • 水文学中,帕累托分布适用于极端事件,例如对于一天内的最大降雨事件,应用软件进行频率分布拟合[32]蓝色图片说明了一个拟合帕累托分布的例子根据二项分布对年最大单日降雨量进行排名,也显示了90%的置信带。降雨数据由绘图位置s表示,作为累积频率分析的一部分。

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)上均匀分布,则可以与(1-U)交换。

Relation to Zipf's law与齐普夫定律的关系

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]\displaystyle{ x }[/math] values (incomes) are binned into [math]\displaystyle{ N }[/math] ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining [math]\displaystyle{ x_m }[/math] so that [math]\displaystyle{ \alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)} }[/math] where [math]\displaystyle{ H(N,\alpha-1) }[/math] 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,有时也称为zeta分布,是一个离散分布,将值分成一个简单的排名。它们都是一个简单的幂律,具有负指数,缩放后它们的累积分布等于1。如果将[math]\displaystyle{ x }[/math]值(收入)组合到[math]\displaystyle{ N }[/math]等级中,那么Zipf可以从Pareto分布中得出,这样每个箱子中的人数遵循1/rank模式。通过定义[math]\displaystyle{ xum }[/math]使[math]\displaystyle{ \alpha x\\mathrm{m}\alpha=\frac{1}{H(N,alpha-1)} }[/math]其中[math]\displaystyle{ H(N,alpha-1) }[/math]广义调和数。这使得Zipf的概率密度函数可以从Pareto得到。

[math]\displaystyle{ f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)} }[/math]


where [math]\displaystyle{ s = \alpha-1 }[/math] and [math]\displaystyle{ 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]\displaystyle{ f(x) }[/math] probability of ranking [math]\displaystyle{ x }[/math].

其中[math]\displaystyle{ s=\alpha-1 }[/math][math]\displaystyle{ x }[/math]是一个整数,表示从1到N的等级,其中N是最高收入等级。因此,从某一人群(或语言、互联网或国家)中随机选择的人(或单词、网站链接或城市)具有[math]\displaystyle{ f(x) }[/math]排名概率[math]\displaystyle{ x }[/math]

Relation to the "Pareto principle"与“帕累托原则”的关系

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]\displaystyle{ \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[33] to be mathematically equivalent:

80-20定律”,根据这个定律,20%的人得到所有收入的80%,而20%最富裕的20%的人得到这80%的80%,依此类推,当帕累托指数为[math]\displaystyle{ \alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161 }[/math]时,这条定律成立。这个结果可以从下面给出的洛伦兹曲线公式中得出。此外,以下内容[34]在数学上是等价的:

  • Income is distributed according to a Pareto distribution with index α > 1.
  • 收入按照指数为α > 1的帕累托分布进行分配。
  • There is some number 0 ≤ p ≤ 1/2 such that 100p % of all people receive 100(1 − p)% of all income, and similarly for every real (not necessarily integer) n > 0, 100pn % of all people receive 100(1 − p)n percentage of all income. α and p are related by
  • 有一个数字0 ≤ p ≤ 1/2,即所有人中的100p %获得全部收入的100(1 − p)%,同样,每个实数(不一定是整数)n > 0,所有人中100pn %得到收入的100(1 − p)n 。”α”和“p”有以下关系:
[math]\displaystyle{ 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.

这不仅适用于收入,也适用于财富,或任何可以用这种分布来模拟的东西。

This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.

这不包括0<“α”≤1的帕累托分布,如前所述,其具有无限的期望值,因此无法合理模拟收入分配。

Relation to Price's law与价格定律的关系

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]\displaystyle{ \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.

Price的平方根定律有时作为帕累托分布的属性或类似于帕累托分布提供。然而,该定律只适用于[math]\displaystyle{ \alpha=1 }[/math]的情况。请注意,在这种情况下,没有定义财富的总量和预期金额,而且该规则只适用于渐近随机样本。上面提到的扩展帕累托原则是一个更一般的规则。

Lorenz curve and Gini coefficient洛伦兹曲线与基尼系数

| title=Ecrits sur la courbe de la répartition de la richesse

|关于财富分配曲线的文章

| title=Ecrits sur la courbe de la répartition de la richesse

|财富分配曲线上的文字

文件:ParetoLorenzSVG.svg
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)

许多帕累托分布的拇指| 325px |洛伦兹曲线。情形“α”==&nbsp;∞对应于完全相等分布(“G”=&nbsp;0),而“α”==1行对应于完全不等式(“G”=&nbsp;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 Lorenz曲线通常用于描述收入和财富分配。对于任何分布,洛伦兹曲线“L”(“F”)用PDF“F”或CDF“F”表示为

| editor=Librairie Droz

编辑: Librairie Droz


| year=1965

1965年

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 0\lt \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. 对于[math]\displaystyle{ 0\lt \alpha\le 1 }[/math]分母是无穷大的,得到“L”=0。右图显示了一些Pareto分布的Lorenz曲线示例。

| 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.[35] We can estimate the Pareto index that would apply to this situation. Letting ε equal [math]\displaystyle{ 62/(7\times 10^9) }[/math] we have:

根据Oxfam(2016年),最富有的62人拥有的财富与世界上最贫穷的一半人口的财富相同。[36]我们可以估计适用于这种情况的帕累托指数。让ε等于[math]\displaystyle{ 62/(7乘以10^9) }[/math]我们得到:

[math]\displaystyle{ L(1/2)=1-L(1-\varepsilon) }[/math]

or

[math]\displaystyle{ 1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}} }[/math]

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.[37]

| 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]\displaystyle{ \alpha\ge 1 }[/math]) to be

| last1=Crovella

1 = Crovella


| author-link1=Mark Crovella

1 = Mark Crovella

[math]\displaystyle{ 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计算方法

| number=6

6

Random sample generation随机样本生成

| 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 on the unit interval (0, 1], the variate T given by

可以使用反变换采样生成随机样本。给定随机变量“U”,从单位间隔(0,1)上的均匀分布中提取,变量“T”由

| date=December 1997}}

| date = December 1997}


[math]\displaystyle{ T=\frac{x_\mathrm{m}}{U^{1/\alpha}} }[/math]


is Pareto-distributed.[38] If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).

是帕累托分布的。[39]如果“U”在[0,1”上均匀分布,则可以与(1-“U”)交换。

See also另请参阅

Category:Actuarial science

类别: 精算

Category:Continuous distributions

类别: 连续分布

Category:Power laws

分类: 权力法则

Category:Probability distributions with non-finite variance

范畴: 非有限方差的概率分布

Category:Exponential family distributions

类别: 指数族分布

Category:Vilfredo Pareto

类别: Vilfredo Pareto


This page was moved from wikipedia:en:Pareto distribution. Its edit history can be viewed at Pareto分布/edithistory

此页摘自维基百科:英语:帕累托分布。编辑历史可在Pareto distribution / edithistory < / small > < / noinclude >查阅。

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  9. 9.0 9.1 Christian Kleiber & Samuel Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 978-0-471-15064-0. https://books.google.com/books?id=7wLGjyB128IC.  引用错误:无效<ref>标签;name属性“kk03”使用不同内容定义了多次
  10. 10.0 10.1 Feller, W. (1971). An Introduction to Probability Theory and its Applications. II (2nd ed.). New York: Wiley. p. 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 ~ Axα as x → ∞."
  11. {cite journal | last1=Lomax | first1=K.S.| year=1954 | title=Business failures。另一个失败数据分析的例子| url=|journal=journal of the American Statistical Association | volume=49 | issue=268 | pages=847–52 | doi=10.1080/01621459.1954.10501239}
  12. 引用错误:无效<ref>标签;未给name属性为Feller的引用提供文字
  13. Chotikapanich, Duangkamon (16 September 2008). "Chapter 7: Pareto and Generalized Pareto Distributions". Modeling Income Distributions and Lorenz Curves. pp. 121–22. ISBN 9780387727967. https://books.google.com/books?id=fUJZZLj1kbwC. 
  14. Chotikapanich, Duangkamon (16 September 2008). "Chapter 7: Pareto and Generalized Pareto Distributions". Modeling Income Distributions and Lorenz Curves. pp. 121–22. ISBN 9780387727967. https://books.google.com/books?id=fUJZZLj1kbwC. 
  15. http://www.cs.bgu.ac.il/~mps042/invtransnote.htm
  16. http://www.cs.bgu.ac.il/~mps042/invtransnote.htm
  17. 17.0 17.1 17.2 17.3 17.4 Huang, Xiao-dong (2004). "A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic". IEEE Transactions on Broadcasting. 50 (3): 323–334. doi:10.1109/TBC.2004.834013.
  18. 18.0 18.1 18.2 18.3 18.4 18.5 Reed, William J.; et al. (2004). "The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions". Communications in Statistics – Theory and Methods. 33 (8): 1733–53. CiteSeerX 10.1.1.70.4555. doi:10.1081/sta-120037438. Unknown parameter |s2cid= ignored (help)
  19. 19.0 19.1 Reed, William J. (2002). "On the rank‐size distribution for human settlements". Journal of Regional Science. 42 (1): 1–17. doi:10.1111/1467-9787.00247. Unknown parameter |s2cid= ignored (help)
  20. 20.0 20.1 20.2 20.3 引用错误:无效<ref>标签;未给name属性为“Reed”的引用提供文字
  21. Schroeder, Bianca; Damouras, Sotirios; Gill, Phillipa (2010-02-24). "Understanding latent sector error and how to protect against them" (PDF). 8th Usenix Conference on File and Storage Technologies (FAST 2010). Retrieved 2010-09-10. 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 (χ2 statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.
  22. Schroeder, Bianca; Damouras, Sotirios; Gill, Phillipa (2010-02-24). "Understanding latent sector error and how to protect against them" (PDF). 8th Usenix Conference on File and Storage Technologies (FAST 2010). Retrieved 2010-09-10. 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 (χ2 statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.
  23. 23.0 23.1 Yuji Ijiri; Simon, Herbert A. (May 1975). "Some Distributions Associated with Bose–Einstein Statistics". Proc. Natl. Acad. Sci. USA. 72 (5): 1654–57. Bibcode:1975PNAS...72.1654I. doi:10.1073/pnas.72.5.1654. PMC 432601. PMID 16578724.
  24. Harchol-Balter, Mor; Downey, Allen (August 1997). "Exploiting Process Lifetime Distributions for Dynamic Load Balancing" (PDF). ACM Transactions on Computer Systems. 15 (3): 253–258. doi:10.1145/263326.263344. Unknown parameter |s2cid= ignored (help)
  25. Harchol-Balter, Mor; Downey, Allen (August 1997). "Exploiting Process Lifetime Distributions for Dynamic Load Balancing" (PDF). ACM Transactions on Computer Systems. 15 (3): 253–258. doi:10.1145/263326.263344. Unknown parameter |s2cid= ignored (help)
  26. 26.0 26.1 [3]
  27. Kleiber and Kotz (2003): p. 94.
  28. Seal, H. (1980). "Survival probabilities based on Pareto claim distributions". ASTIN Bulletin. 11: 61–71. doi:10.1017/S0515036100006620.
  29. Kleiber and Kotz (2003): p. 94.
  30. Seal, H. (1980). "Survival probabilities based on Pareto claim distributions". ASTIN Bulletin. 11: 61–71. doi:10.1017/S0515036100006620.
  31. CumFreq, software for cumulative frequency analysis and probability distribution fitting [4]
  32. CumFreq, software for cumulative frequency analysis and probability distribution fitting [5]
  33. Hardy, Michael (2010). "Pareto's Law". Mathematical Intelligencer. 32 (3): 38–43. doi:10.1007/s00283-010-9159-2. Unknown parameter |s2cid= ignored (help)
  34. Hardy, Michael (2010). "Pareto's Law". Mathematical Intelligencer. 32 (3): 38–43. doi:10.1007/s00283-010-9159-2. Unknown parameter |s2cid= ignored (help)
  35. "62 people own the same as half the world, reveals Oxfam Davos report". Oxfam. Jan 2016.
  36. {cite web | title=62人拥有的财富与世界上一半的人相同,展示乐施会达沃斯报告网址=https://www.oxfam.org/en/pressroom/pressreases/2016-01-18/62-people-own-same-half-world-reviews-oxfam-davos-report%7Cpublisher=Oxfam | date=2016年1月}
  37. "Global Wealth Report 2013". Credit Suisse. Oct 2013. p. 22. Archived from the original on 2015-02-14. Retrieved 2016-01-24.
  38. Tanizaki, Hisashi (2004). Computational Methods in Statistics and Econometrics. CRC Press. p. 133. ISBN 9780824750886. https://books.google.com/books?id=pOGAUcn13fMC. 
  39. {cite book | last=Tanizaki | first=Hisashi | title=统计学和计量经济学中的计算方法| year=2004 | page=133 | publisher=CRC Press |网址=https://books.google.com/books?id=pOGAUcn13fMC | isbn=9780824750886}