User:Mathstat/Pareto generalizations

The Generalized Pareto Distributions

Note: Most of the material below has been added to Pareto distribution. See the current revision of the article Pareto distribution and the Talk page Talk:Pareto distribution.

There is a hierarchy of Pareto Distributions known as Pareto Type I, II, III, IV, and Feller-Pareto distributions. Pareto Type IV contains Pareto Type I and II as special cases. The Feller-Pareto distribution generalizes Pareto Type IV.

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,

Pareto Distributions

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

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:

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

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

Feller-Pareto distribution

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

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

where B( ) is the beta function.

When $\gamma _{2}=1$ W has the Lomax distribution, and $\mu +x_{m}W$ 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.

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

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

References

• N. L. Johnson, S. Kotz, and N. Balakrishnan (1994). Continuous Univariate Distributions Volume 1, Second Edition, Wiley.
• Barry C. Arnold (2011). "Chapter 7: Pareto and Generalized Pareto Distributions". In Duangkamon Chotikapanich (ed.). Modeling Distributions and Lorenz Curves. New York: Springer. ISBN 9780387727967. {{cite book}}: Text "." ignored (help)
• Arnold, B. C. and Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics. {{cite book}}: Text "." ignored (help)CS1 maint: multiple names: authors list (link)