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,

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:


 * $$ P(IV)(x_m, x_m, 1, \alpha) = P(I)(x_m, \alpha),$$ 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 00,

$$ 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.