# 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 [1][2] of Pareto Distributions known as Pareto Type I, II, III, IV, and Feller-Pareto distributions.[3] Pareto Type IV contains Pareto Type I and II as special cases. The Feller-Pareto[4][2] 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,[5]

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 ${\displaystyle \left[1+\left({\frac {x-\mu }{x_{m}}}\right)^{1/\gamma }\right]^{-\alpha }}$ ${\displaystyle x>\mu }$ ${\displaystyle \alpha ,x_{m}>0,\mu \in \mathbb {R} }$

The shape parameter α is the tail index, μ is location, xm 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

${\displaystyle f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 00,}$

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.

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 ${\displaystyle {\frac {\sigma \Gamma (\alpha -\gamma \alpha )\Gamma (1+\gamma )}{\Gamma (\alpha )}}}$ ${\displaystyle -1<\gamma <\alpha }$ ${\displaystyle {\frac {\sigma ^{\delta }\Gamma (\alpha -\gamma \alpha )\Gamma (1+\gamma \delta )}{\Gamma (\alpha )}}}$ ${\displaystyle -\gamma ^{-1}<\delta <\alpha /\gamma }$

## Notes

1. ^ Arnold (1983), p. 45 (3.2.2).
2. ^ a b c Johnson, Kotz, and Balakrishnan (1994), page 575, (20.4). Cite error: Invalid <ref> tag; name "jkb94" defined multiple times with different content (see the help page).
3. ^ See Johnson, Kotz, and Balakrishnan (1994), Ch. 20, Arnold (1983), Ch. 3, and Kleiber and Kotz (2003), Ch. 3.
4. ^ Feller, W. (1971). An Introduction to Probability Theory and its Applications, 2 (Second edition), New York: Wiley.
5. ^ Lomax, K. S. (1954). Business failures. Another example of the analysis of failure data. Journal of the American Statistical Association, 49, 847–852.
6. ^ Feller, W. (1971). An Introduction to Probability Theory and its Applications, 2 (Second edition), New York: Wiley.

## 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. Modeling Distributions and Lorenz Curves. New York: Springer. 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. Text "." ignored (help)