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第422行： + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − | Type II+ − | 第二类
→Pareto types I–IV
====Pareto types I–IV====
====Pareto types I–IV====
{|class="wikitable" border="1"
|+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.
The Pareto distribution hierarchy is summarized in the next table comparing the [[survival function]]s (complementary CDF).
The Pareto distribution hierarchy is summarized in the next table comparing the [[survival function]]s (complementary CDF).