Pareto分布

帕累托分布 Pareto distribution，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，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分布。其最初用于描述一个社会中的财富分配，拟合大部分财富由一小部分人口持有的分布情况， 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”。例如一个社会80% 的财富掌握在20% 的人口手中这样的社会现象。然而，帕累托分布与帕雷托法则并不能混为一谈，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。

定义

如果 X 是一个具有 帕累托分布 （I 型分布）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 Survival function (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} \left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\ 1 & x \lt x_\mathrm{m}, \end{cases} }[/math]

其中，x_m是“X”的最小可能值（必为正），“α”是一个正参数。帕累托 I 型分布由比例参数x_m和形状参数“α”所决定，即所谓的“尾部指数 tail index”。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Pareto index。

Cumulative distribution function 累积分布函数

根据定义，带有参数 α 和 x_m 的 Pareto 随机变量的累积分布函数Cumulative distribution function 是

[math]\displaystyle{ F_X(x) = \begin{cases} 1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\ 0 & x \lt x_\mathrm{m}. \end{cases} }[/math]

Probability density function概率密度函数

由此可以得出概率密度函数Probability density function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]

当在线性轴上绘制时，我们默认分布曲线为我们熟知的的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log-log plot中绘制时，分布则是直线表示。

Properties性质

Moments and characteristic function矩与特征函数

• 服从帕累托分布的随机变量的期望值expected value为：

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

• 服从帕累托分布的随机变量的方差variance为

[math]\displaystyle{ \operatorname{Var}(X)= \begin{cases} \infty & \alpha\in(1,2], \\ \left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha\gt 2. \end{cases} }[/math]

(如果 α ≤1，方差不存在.)

• 原始的矩raw moments为：

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

• 其力矩生成函数moment generating function仅对非正值t≤0有定义：

[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]

[math]\displaystyle{ M\left(0,\alpha,x_\mathrm{m}\right)=1. }[/math]

由于其期望不会收敛到特定区间，包括t=0的情况，于是我们称其力矩生成函数并不存在。

• 特征函数characteristic function由以下给出

[math]\displaystyle{ \varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t), }[/math]

其中 Γ(a, x) 是不完全伽马函数incomplete gamma function, 其系数可能被矩量法method of moments来解。

Conditional distributions条件分布

对于一个符合Pareto分布随机变量的大于或等于某个特定的数[math]\displaystyle{ x_1 }[/math] > [math]\displaystyle{ x_\text{m} }[/math]的条件概率分布，其就是具有相同的帕累托指数的帕累托分布但是其最小值为[math]\displaystyle{ x_1 }[/math]而不是[math]\displaystyle{ x_\text{m} }[/math]。

A characterization theorem一个特征定理

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，[math]\displaystyle{ x_\text{m}\gt 0 }[/math]。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，那么其共同分布就是帕累托分布。

Geometric mean几何平均数

The geometric mean (G) is

几何平均数 geometric mean (G)是：

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

Harmonic mean调和平均数

调何平均数 harmonic mean (H) 有：

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

Graphical representation图示法

当在线性轴上绘制时，分布曲线为我们熟知的的J形曲线，该曲线渐近地接近每个正交轴。在双对数图log-log plot中绘制时，分布则是直线表示，即对于x ≥ x_m的概率密度函数。

[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]

由于α在公式中为正，因此梯度−(α + 1)为负。

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。

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。

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)。

帕累托分布的调何平均数(H)有帕累托 I 型、 II 型、 III 型、 IV 型和 Feller-Pareto分布。帕累托 IV 型是帕累托类型 I-III 中的特殊情况。Feller-Pareto分布推广了帕累托 IV 型。

Related distributions相关分布

Generalized Pareto distributions广义帕累托分布

There is a hierarchy 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. 帕累托分布有一个层次结构帕累托分布被称为帕累托类型I、II、III、IV，费勒-帕累托分布。Pareto类型IV包含Pareto类型I-III作为特殊情况。费勒-帕累托

如果[math]\displaystyle{ U_1 \sim \Gamma(\delta_1, 1) }[/math] 和 [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

\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 是 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 x_m 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x_m 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

是指数分布，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 x_m 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  =

��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，分别服从指数分布和正态分布。（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] 和 [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 =

See also: Truncated distribution

| 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.

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

概率密度函数是

Therefore, the logarithmic likelihood function is

因此， 对数似然函数 Logarithmic Likelihood function是

[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

如果“U”在（0，1）上为均匀分布，则应用反变换方法

\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. 是一个有界的帕累托分布。

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.

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: 对称帕累托分布的累积分布函数（CDF）定义如下：

\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:

相应的概率密度函数（PDF）为：

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：

• 频率名-->

\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)

• 人类居住区的规模（少数城市，许多村庄/村庄）

-->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)

• 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）

• Hard disk drive error rates

• 硬盘驱动器错误率

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
• 绝对零度附近的玻色—爱因斯坦凝聚 簇

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 values of oil reserves in oil fields (a few large fields, many small fields)
• 油田的石油储量值（少数大油田，许多小油田）

• The length distribution in jobs assigned to supercomputers (a few large ones, many small ones)

• 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）

(see Aaberge 2005).

(见 Aaberge 2005)。

• The standardized price returns on individual stocks

• 个股的标准化价格回报率

• Sizes of sand particles

• 沙粒尺寸

• The size of meteorites

• 陨石的大小

• Male dating success on Tinder [80% of females compete for the 20% most attractive males]

• 在Tinder上，男性约会成功【80%的女性竞争20%最具吸引力的男性】

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.

• 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，

• 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 }}

• In hydrology the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. 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 positions as part of the cumulative frequency analysis.

• 在水文学中，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]是广义调和数。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]和[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 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. 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。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.

| 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. 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。

See also另请参阅

• Bradford's law
• 布拉德福德定律

• Gutenberg–Richter law

• 古腾堡-里克特定律

• Matthew effect

• 马太效应

Category:Actuarial science

类别: 精算

• Pareto analysis

• 帕累托分析

Category:Continuous distributions

类别: 连续分布

• Pareto efficiency

• 帕累托效率

Category:Power laws

分类: 权力法则

• Pareto interpolation

• 帕累托插值

Category:Probability distributions with non-finite variance

范畴: 非有限方差的概率分布

• Power law probability distributions

• 幂律概率分布

Category:Exponential family distributions

类别: 指数族分布

• Sturgeon's law

• 鲟鱼定律

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。