User:Mathstat/Pareto generalizations

Pareto Types I-IV

The Pareto distribution hierarchy is summarized in the table comparing the survival distributions (complementary CDF). The Pareto distribution of the second kind is also known as the Lomax distribution,^[5]

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Pareto Distributions

	${\displaystyle {\overline {F}}(x)=1-F(x)}$	Support	Parameters
Type I	${\displaystyle \left[{\frac {x}{x_{m}}}\right]^{-\alpha }}$	${\displaystyle x>x_{m}}$	${\displaystyle \alpha ,x_{m}>0}$
Type II	${\displaystyle \left[1+{\frac {x-\mu }{x_{m}}}\right]^{-\alpha }}$	${\displaystyle x>\mu }$	${\displaystyle \alpha ,x_{m}>0,\mu \in \mathbb {R} }$
Lomax	${\displaystyle {\frac {C^{\alpha }}{(x+C)^{\alpha }}}}$	${\displaystyle x\geq 0}$	${\displaystyle \alpha ,C>0}$
Type III	${\displaystyle \left[1+\left({\frac {x-\mu }{x_{m}}}\right)^{1/\gamma }\right]^{-1}}$	${\displaystyle x>\mu }$	${\displaystyle \alpha ,x_{m}>0,\mu \in \mathbb {R} ,\gamma >0}$
Type IV	${\displaystyle \left[1+\left({\frac {x-\mu }{x_{m}}}\right)^{1/\gamma }\right]^{-\alpha }}$	${\displaystyle x>\mu }$	${\displaystyle \alpha ,x_{m}>0,\mu \in \mathbb {R} }$

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The shape parameter α is the tail index, μ is location, x_m is scale, 'γ is an inequality parameter. Some special cases of Pareto Type (IV) are:

${\displaystyle P(IV)(x_{m},x_{m},1,\alpha )=P(I)(x_{m},\alpha ),}$ and

${\displaystyle P(IV)(\mu ,x_{m},1,\alpha )=P(II)(\mu ,x_{m},\alpha ).}$

Feller-Pareto distribution

Feller^[6][2] defines a Pareto variable by transformation ${\displaystyle W=Y^{-1}-1}$ of a beta random variable Y, where the probability density function of Y is

${\displaystyle f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 0<y<1;\gamma _{1},\gamma _{2}>0,}$

where B( ) is the beta function.

When ${\displaystyle \gamma _{2}=1}$ W has the Lomax distribution, and ${\displaystyle \mu +x_{m}W}$ is a generalization of P(IV).

Properties

Existence of the mean, and variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments exist for some δ>0, as shown in the table below, where δ is not necessarily an integer.

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Moments of Pareto I-IV Distributions (case μ=0)

	${\displaystyle E[X]}$	Condition	${\displaystyle E[X^{\delta }]}$	Condition
Type I	${\displaystyle {\frac {\sigma \alpha }{\alpha -1}}}$	${\displaystyle \alpha >1}$	${\displaystyle {\frac {\sigma ^{\delta }\alpha }{\alpha -\delta }}}$	${\displaystyle \delta <\alpha }$
Type II	${\displaystyle {\frac {\sigma }{\alpha -1}}}$	${\displaystyle \alpha >1}$	${\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\delta )\Gamma (1+\delta )}{\Gamma (\alpha )}}}$	${\displaystyle -1<\delta <\alpha }$
Type III	${\displaystyle \sigma \Gamma (1-\gamma )\Gamma (1+\gamma )}$	${\displaystyle -1<\gamma <1}$	${\displaystyle \sigma ^{\delta }\Gamma (1-\gamma \delta )\Gamma (1+\gamma \delta )}$	${\displaystyle -\gamma ^{-1}<\delta <\gamma ^{-1}}$
Type IV	${\displaystyle {\frac {\sigma \Gamma (\alpha -\gamma \alpha )\Gamma (1+\gamma )}{\Gamma (\alpha )}}}$	${\displaystyle -1<\gamma <\alpha }$	${\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\gamma \alpha )\Gamma (1+\gamma \delta )}{\Gamma (\alpha )}}}$	${\displaystyle -\gamma ^{-1}<\delta <\alpha /\gamma }$

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