Editing Pareto distribution (section)

==Related distributions==
===Generalized Pareto distributions===
{{See also|Generalized Pareto distribution}}

There is a hierarchy <ref name=arnold/><ref name=jkb94>Johnson, Kotz, and Balakrishnan (1994), (20.4).</ref> of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.<ref name=arnold/><ref name=jkb94/><ref name=kk03>{{cite book |author1=Christian Kleiber  |author2=Samuel Kotz  |name-list-style=amp |year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences |publisher=[[John Wiley & Sons|Wiley]] |isbn=978-0-471-15064-0| url=https://books.google.com/books?id=7wLGjyB128IC}}</ref> Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto<ref name=jkb94/><ref name=feller>{{cite book|last=Feller |first= W.| year=1971| title=An Introduction to Probability Theory and its Applications| volume=II| edition=2nd | location= New York|publisher=Wiley|page=50}} "The densities (4.3) are sometimes called after the economist ''Pareto''. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ ''Ax''<sup>−''α''</sup> as ''x''&nbsp;→&nbsp;∞".</ref> distribution generalizes Pareto Type IV.
<!--- In this context using x_m for the lower bound for the scale parameter is not meaningful, usual notation is \sigma --->

====Pareto types I–IV====
The Pareto distribution hierarchy is summarized in the next table comparing the [[survival function]]s (complementary CDF).

When ''μ'' = 0, the Pareto distribution Type II is also known as the [[Lomax distribution]].<ref>{{cite journal | last1 = Lomax | first1 = K. S. | year = 1954 | title = Business failures. Another example of the analysis of failure data | journal = Journal of the American Statistical Association | volume = 49 | issue = 268| pages = 847–52 | doi=10.1080/01621459.1954.10501239}}</ref>

In this section, the symbol ''x''<sub>m</sub>, used before to indicate the minimum value of ''x'', is replaced by&nbsp;''σ''.

{|class="wikitable"
|+Pareto distributions
! !! <math> \overline{F}(x)=1-F(x)</math> !! Support !! Parameters
|-
| Type I
|| <math>\left[\frac x \sigma \right]^{-\alpha}</math>
|| <math>x \ge \sigma</math>
|| <math>\sigma > 0, \alpha</math>
|-
| Type II
|| <math>\left[1 + \frac{x-\mu} \sigma \right]^{-\alpha}</math>
|| <math>x \ge \mu</math>
|| <math>\mu \in \mathbb R, \sigma > 0, \alpha</math>
|-
| Lomax
|| <math>\left[1 + \frac x \sigma \right]^{-\alpha}</math>
|| <math>x \ge 0</math>
|| <math>\sigma > 0, \alpha</math>
|-
| Type III
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1} </math>
|| <math>x \ge \mu</math>
|| <math> \mu \in \mathbb R, \sigma, \gamma > 0</math>
|-
| Type IV
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha}</math>
|| <math>x \ge \mu</math>
|| <math>\mu \in \mathbb R, \sigma, \gamma > 0, \alpha</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

::<math> P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),</math>
::<math> P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),</math>
::<math> P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).</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.

{|class="wikitable"
|+Moments of Pareto I–IV distributions (case ''μ'' = 0)
! !! <math>\operatorname{E}[X]</math> !! Condition !! <math>\operatorname{E}[X^\delta]</math> !! Condition
|-
| Type I
|| <math>\frac{\sigma \alpha}{\alpha-1}</math>
|| <math>\alpha > 1</math>
|| <math>\frac{\sigma^\delta \alpha}{\alpha-\delta}</math>
|| <math> \delta < \alpha</math>
|-
| Type II
|| <math> \frac{ \sigma }{\alpha-1}+\mu</math>
|| <math>\alpha > 1</math>
|| <math> \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}</math>
|| <math>0 < \delta < \alpha</math>
|-
| Type III
|| <math>\sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)</math>
|| <math> -1<\gamma<1</math>
|| <math>\sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)</math>
|| <math>-\gamma^{-1}<\delta<\gamma^{-1}</math>
|-
| Type IV
|| <math>\frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)}</math>
|| <math> -1<\gamma<\alpha</math>
|| <math>\frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}</math>
|| <math>-\gamma^{-1}<\delta<\alpha/\gamma </math>
|-
|-
|}

====Feller–Pareto distribution====
Feller<ref name=jkb94/><ref name=feller/> defines a Pareto variable by transformation ''U''&nbsp;=&nbsp;''Y''<sup>−1</sup>&nbsp;−&nbsp;1 of a [[beta distribution|beta random variable]] ,''Y'', whose probability density function is

:<math> f(y) = \frac{y^{\gamma_1-1} (1-y)^{\gamma_2-1}}{B(\gamma_1, \gamma_2)}, \qquad 0<y<1; \gamma_1,\gamma_2>0,</math>

where ''B''(&nbsp;) is the [[beta function]]. If

:<math> W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma>0, \gamma>0,</math>

then ''W'' has a Feller–Pareto distribution FP(''μ'', ''σ'', ''γ'', ''γ''<sub>1</sub>, ''γ''<sub>2</sub>).<ref name=arnold/>

If <math>U_1 \sim \Gamma(\delta_1, 1)</math> and <math>U_2 \sim \Gamma(\delta_2, 1)</math> are independent [[Gamma distribution|Gamma variables]], another construction of a Feller–Pareto (FP) variable is<ref>{{cite book |last=Chotikapanich |first=Duangkamon |title=Modeling Income Distributions and Lorenz Curves |chapter=Chapter 7: Pareto and Generalized Pareto Distributions |date=16 September 2008 |pages=121–22 |publisher=Springer |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}</ref>

:<math>W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma</math>

and we write ''W'' ~ FP(''μ'', ''σ'', ''γ'', ''δ''<sub>1</sub>, ''δ''<sub>2</sub>). Special cases of the Feller–Pareto distribution are

:<math>FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)</math>
:<math>FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)</math>
:<math>FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)</math>
:<math>FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).</math>

===Inverse-Pareto Distribution / Power Distribution ===

When a random variable <math>Y</math> follows a pareto distribution, then its inverse <math>X=1/Y</math> follows a Power distribution. 
Inverse Pareto distribution is equivalent to a Power distribution <ref>Dallas, A. C. "Characterizing the Pareto and power distributions." Annals of the Institute of Statistical Mathematics 28.1 (1976): 491-497.</ref>

:<math>Y\sim \mathrm{Pa}(\alpha, x_m) = \frac{\alpha x_m^\alpha}{y^{\alpha+1}} \quad (y \ge x_m) \quad \Leftrightarrow \quad X\sim \mathrm{iPa}(\alpha, x_m) = \mathrm{Power}(x_m^{-1}, \alpha) = \frac{\alpha x^{\alpha-1}}{(x_m^{-1})^\alpha} \quad (0< x \le x_m^{-1})</math>

===Relation to the exponential distribution===
The Pareto distribution is related to the [[exponential distribution]] as follows. If ''X'' is Pareto-distributed with minimum ''x''<sub>m</sub> and index&nbsp;''α'', then

: <math> Y = \log\left(\frac{X}{x_\mathrm{m}}\right) </math>

is [[exponential distribution|exponentially distributed]] with rate parameter&nbsp;''α''. Equivalently, if ''Y'' is exponentially distributed with rate&nbsp;''α'', then

: <math> x_\mathrm{m} e^Y</math>

is Pareto-distributed with minimum ''x''<sub>m</sub> and index&nbsp;''α''.

This can be shown using the standard change-of-variable techniques:

: <math>
\begin{align}
\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}.
\end{align}
</math>

The last expression is the cumulative distribution function of an exponential distribution with rate&nbsp;''α''.

Pareto distribution can be constructed by hierarchical exponential distributions.<ref>{{Cite thesis|title=Bayesian semiparametric spatial and joint spatio-temporal modeling|url=https://mospace.umsystem.edu/xmlui/handle/10355/4450|publisher=University of Missouri--Columbia|date=2006|degree=Thesis|first=Gentry|last=White}} section 5.3.1.</ref> Let
<math>\phi | a \sim \text{Exp}(a)</math> and
<math>\eta | \phi \sim \text{Exp}(\phi) </math>. Then we have <math>p(\eta | a) = \frac{a}{(a+\eta)^2}</math> and, as a result, <math>a+\eta \sim \text{Pareto}(a, 1)</math>.

More in general, if <math>\lambda \sim \text{Gamma}(\alpha, \beta)</math> (shape-rate parametrization) and <math>\eta | \lambda \sim \text{Exp}(\lambda) </math>, then <math>\beta + \eta \sim \text{Pareto}(\beta, \alpha)</math>.

Equivalently, if <math>Y \sim \text{Gamma}(\alpha,1) </math> and <math>X \sim \text{Exp}(1)</math>, then <math>x_{\text{m}} \! \left(1 + \frac{X}{Y}\right) \sim \text{Pareto}(x_{\text{m}}, \alpha)</math>.

===Relation to the log-normal distribution===
The Pareto distribution and [[log-normal distribution]] are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the [[exponential distribution]] and [[normal distribution]]. (See [[#Relation_to_the_exponential_distribution|the previous section]].)

===Relation to the generalized Pareto distribution===
The Pareto distribution is a special case of the [[generalized Pareto distribution]], which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the [[Lomax distribution]] as a special case. This family also contains both the unshifted and shifted [[exponential distribution]]s.

The Pareto distribution with scale <math>x_m</math> and shape <math>\alpha</math> is equivalent to the generalized Pareto distribution with location <math>\mu=x_m</math>, scale <math>\sigma=x_m/\alpha</math> and shape <math>\xi=1/\alpha</math>  and, conversely, one can get the Pareto distribution from the GPD by taking <math>x_m = \sigma/\xi</math> and <math>\alpha=1/\xi</math> if <math>\xi > 0</math>.

===Bounded Pareto distribution===
{{See also|Truncated distribution}}
{{Probability distribution
 | name       =Bounded Pareto
 | type       =density
 | pdf_image  =
 | cdf_image  =
 | parameters =
<math>L > 0</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>H > L</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>\alpha > 0</math> [[shape parameter|shape]] (real)
 | support    =<math>L \leqslant x \leqslant H</math>
 | pdf        =<math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
 | cdf        =<math>\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
 | mean       =
<math>\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1 </math><br />
<math>\frac{{H}{L}}{{H}-{L}}\ln\frac{H}{L}, \alpha=1</math>
 | median     =<math> L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}</math>
 | mode       =
 | variance   =
<math>\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2</math>
<math>\frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2</math>
(this is the second raw moment, not the variance)
 | skewness   = <math>\frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j </math>
(this is the kth raw moment, not the skewness)
 | kurtosis   =
 | entropy    =
 | mgf        =
 | char       =
}}

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 [[probability density function]] is

: <math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>,

where ''L''&nbsp;≤&nbsp;''x''&nbsp;≤&nbsp;''H'', and ''α''&nbsp;>&nbsp;0.

====Generating bounded Pareto random variables====
If ''U'' is [[uniform distribution (continuous)|uniformly distributed]] on (0,&nbsp;1), then applying inverse-transform method <ref>{{Cite web |url=http://www.cs.bgu.ac.il/~mps042/invtransnote.htm |title=Inverse Transform Method |access-date=2012-08-27 |archive-date=2012-01-17 |archive-url=https://web.archive.org/web/20120117042753/http://www.cs.bgu.ac.il/~mps042/invtransnote.htm |url-status=dead }}</ref>

:<math>U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}</math>
:<math>x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}</math>

is a bounded Pareto-distributed.
{{Clear}}

===Symmetric Pareto distribution===
The purpose of the Symmetric and Zero Symmetric Pareto distributions is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from the Pareto distribution. Long probability tails normally means that probability decays slowly, and can be used to fit a variety of datasets. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.<ref name=":0">{{Cite journal|last=Huang|first=Xiao-dong|date=2004|title=A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic|journal=IEEE Transactions on Broadcasting|volume=50|issue=3|pages=323–334|doi=10.1109/TBC.2004.834013}}</ref>

The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:<ref name=":0" />

<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over 2b-X}) ^a  & X<b \\
1- \tfrac{1}{2}(\tfrac{b}{X})^a& X\geq b
\end{cases}</math>

The corresponding probability density function (PDF) is:<ref name=":0" />

<math>p(x) = {ab^a \over 2(b+\left\vert x-b \right\vert)^{a+1}},X\in R</math>

This distribution has two parameters: a and b. It is symmetric about b. Then the mathematic expectation is b. When, it has variance as following:

<math>E((x-b)^2)=\int_{-\infty}^{\infty} (x-b)^2p(x)dx={2b^2 \over (a-2)(a-1) }

</math>

The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:

<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over b-X}) ^a  & X<0 \\
1- \tfrac{1}{2}(\tfrac{b}{b+X})^a& X\geq 0
\end{cases}</math>

The corresponding PDF is:

<math>p(x) = {ab^a \over 2(b+\left\vert x \right\vert)^{a+1}},X\in R</math>

This distribution is symmetric about zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.<ref name=":0" />

===Multivariate Pareto distribution===
The univariate Pareto distribution has been extended to a [[multivariate Pareto distribution]].<ref>{{cite journal
|last1=Rootzén|first1=Holger |last2=Tajvidi|first2=Nader 
|title=Multivariate generalized Pareto distributions |journal=Bernoulli|volume=12|year=2006|number=5 |pages=917–30 
|doi=10.3150/bj/1161614952 
|citeseerx=10.1.1.145.2991|s2cid=16504396 }}</ref>