Editing Pareto distribution (section)

===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>