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第1,141行：
第1,141行： − + 第1,148行：
第1,148行： − + 第1,154行：
第1,154行： − + 第1,160行：
第1,160行： − + 第1,166行：
第1,166行： − + 第1,176行：
第1,176行： − + − 概率密度函数+ 第1,239行：
第1,239行： − 对称帕累托分布和零对称帕累托分布的目的是捕捉一些具有尖锐概率峰值和对称长概率尾的特殊统计分布。这两种分布是从帕累托分布中推导出来的。长概率尾通常意味着概率衰减缓慢。在很多情况下，帕累托分布的工作都很合适。但是，如果分布具有两条慢衰减尾的对称结构，帕累托不能做到这一点。然后应用对称帕累托或零对称帕累托分布。+ − + − 对称累积分布函数帕累托分布系统(CDF)定义如下:+ 第1,261行：
第1,261行： − + 第1,277行：
第1,277行： − 因此，对数似然函数是+ 第1,289行：
第1,289行： − + 第1,295行：
第1,295行： − + + + 第1,320行：
第1,322行： − + 第1,326行：
第1,328行： − −
→Bounded Pareto distribution
峰度 =
峰度 =
===Bounded Pareto distribution===
===Bounded Pareto distribution有界帕累托分布===
| entropy =
| entropy =
{{See also|Truncated distribution}}
{{See also|Truncated distribution}}
{{另请参阅|截断分布}}
| mgf =
| mgf =
{{Probability distribution
{{Probability distribution
{{概率分布
| char =
| char =
| name =Bounded Pareto
| name =Bounded Pareto
名称=有界帕累托
}}
}}
| type =density
| type =density
类型=密度
| pdf_image =
| pdf_image =
| parameters =
| parameters =
参数=
The probability density function is
The probability density function is
概率密度函数为
<math>L > 0</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>L > 0</math> [[location parameter|location]] ([[real numbers|real]])<br />
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.
The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.
对称帕累托分布和零对称帕累托分布的目的是捕捉一些具有尖锐概率峰值和对称长概率尾的特殊统计分布。这两种分布是从帕累托分布中推导出来的。长概率尾通常意味着概率衰减缓慢。在很多情况下，帕累托分布的工作都很合适。但是，如果分布具有两条慢衰减尾的对称结构，帕累托不能做到这一点。然后以应用'''<font color="#ff8000"> 对称帕累托Symmetric Pareto</font>'''或'''<font color="#ff8000"> 零对称帕累托Zero Symmetric Pareto</font>'''来取代。
(this is the kth raw moment, not the skewness)
(this is the kth raw moment, not the skewness)
（这是第k个原始时刻，不是偏斜）
| kurtosis =
| kurtosis =
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:
'''<font color="#ff8000"> 对称累积分布函数帕累托Cumulative distribution function (CDF)</font>'''分布系统定义如下:
| entropy =
| entropy =
The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is
The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is
给定一个独立样本 x = (x1，x2，... ，xn) ，帕累托分布参数 α 和 xm 的似然函数为
给定一个独立样本 x = (x1, x2, ..., xn)，帕累托分布参数 α 和 xm 的'''<font color="#ff8000"> 似然函数Likelihood function</font>'''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.
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.
Therefore, the logarithmic likelihood function is
Therefore, the logarithmic likelihood function is
因此，'''<font color="#ff8000"> 对数似然函数 Logarithmic Likelihood function</font>'''是
: <math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>,
: <math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>,
where ''L'' ≤ ''x'' ≤ ''H'', and ''α'' > 0.
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
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:
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====
====Generating bounded Pareto random variables生成有界Pareto随机变量====
If ''U'' is [[uniform distribution (continuous)|uniformly distributed]] on (0, 1), then applying inverse-transform method <ref>http://www.cs.bgu.ac.il/~mps042/invtransnote.htm</ref>
If ''U'' is [[uniform distribution (continuous)|uniformly distributed]] on (0, 1), then applying inverse-transform method <ref>http://www.cs.bgu.ac.il/~mps042/invtransnote.htm</ref>
如果“U”在（0，1）上为[[均匀分布（连续）|均匀分布]]，则应用反变换方法<ref>http://www.cs.bgu.ac.il/~mps042/invtransnote.htm</ref>
\widehat x_\mathrm{m} = \min_i {x_i}.
\widehat x_\mathrm{m} = \min_i {x_i}.
is a bounded Pareto-distributed.{{Citation needed|date=February 2011}}
is a bounded Pareto-distributed.{{Citation needed|date=February 2011}}
是一个有界的帕累托分布。{{Citation needed|date=February 2011}}
{{Clear}}
{{Clear}}
因此，α 的最大似然估计量是:
因此，α 的最大似然估计量是:
===Symmetric Pareto distribution===
===Symmetric Pareto distribution===