Editing Pareto distribution (section)

====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>
|-
|-
|}