Multivariate Pareto distribution

In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution. There are several different types of univariate Pareto distributions including Pareto Types I−IV and Feller−Pareto. Multivariate Pareto distributions have been defined for many of these types.

Bivariate Pareto distributions

Bivariate Pareto distribution of the first kind

Mardia (1962) defined a bivariate distribution with cumulative distribution function (CDF) given by

[math]\displaystyle{ F(x_1, x_2) = 1 -\sum_{i=1}^2\left(\frac{x_i}{\theta_i}\right)^{-a}+ \left(\sum_{i=1}^2 \frac{x_i}{\theta_i} - 1\right)^{-a}, \qquad x_i \gt \theta_i \gt 0, i=1,2; a\gt 0, }[/math]

and joint density function

[math]\displaystyle{ f(x_1, x_2) = (a+1)a(\theta_1 \theta_2)^{a+1}(\theta_2x_1 + \theta_1x_2 - \theta_1 \theta_2)^{-(a+2)}, \qquad x_i \geq \theta_i\gt 0, i=1,2; a\gt 0. }[/math]

The marginal distributions are Pareto Type 1 with density functions

[math]\displaystyle{ f(x_i)=a\theta_i^a x_i^{-(a+1)}, \qquad x_i \geq \theta_i\gt 0, i=1,2. }[/math]

The means and variances of the marginal distributions are

[math]\displaystyle{ E[X_i] = \frac{a \theta_i}{a-1}, a\gt 1; \quad Var(X_i)=\frac{a\theta_i^2}{(a-1)^2(a-2)}, a\gt 2; \quad i=1,2, }[/math]

and for a > 2, X₁ and X₂ are positively correlated with

[math]\displaystyle{ \operatorname{cov}(X_1, X_2) = \frac{\theta_1 \theta_2}{(a-1)^2 (a-2)}, \text{ and } \operatorname{cor}(X_1, X_2) = \frac{1}{a}. }[/math]

Bivariate Pareto distribution of the second kind

Arnold suggests representing the bivariate Pareto Type I complementary CDF by

[math]\displaystyle{ \overline{F}(x_1,x_2) = \left(1 + \sum_{i=1}^2 \frac{x_i-\theta_i}{\theta_i} \right)^{-a}, \qquad x_i \gt \theta_i, i=1,2. }[/math]

If the location and scale parameter are allowed to differ, the complementary CDF is

[math]\displaystyle{ \overline{F}(x_1,x_2) = \left(1 + \sum_{i=1}^2 \frac{x_i-\mu_i}{\sigma_i} \right)^{-a}, \qquad x_i \gt \mu_i, i=1,2, }[/math]

which has Pareto Type II univariate marginal distributions. This distribution is called a multivariate Pareto distribution of type II by Arnold. (This definition is not equivalent to Mardia's bivariate Pareto distribution of the second kind.)

For a > 1, the marginal means are

[math]\displaystyle{ E[X_i] = \mu_i + \frac{\sigma_i}{a-1}, \qquad i=1,2, }[/math]

while for a > 2, the variances, covariance, and correlation are the same as for multivariate Pareto of the first kind.

Multivariate Pareto distributions

Multivariate Pareto distribution of the first kind

Mardia's Multivariate Pareto distribution of the First Kind has the joint probability density function given by

[math]\displaystyle{ f(x_1,\dots,x_k) = a(a+1)\cdots(a+k-1) \left(\prod_{i=1}^k \theta_i \right)^{-1} \left(\sum_{i=1}^k \frac{x_i}{\theta_i} - k + 1 \right)^{-(a+k)}, \qquad x_i \gt \theta_i \gt 0, a \gt 0, \qquad (1) }[/math]

The marginal distributions have the same form as (1), and the one-dimensional marginal distributions have a Pareto Type I distribution. The complementary CDF is

[math]\displaystyle{ \overline{F}(x_1,\dots,x_k) = \left(\sum_{i=1}^k \frac{x_i}{\theta_i}-k+1 \right)^{-a}, \qquad x_i \gt \theta_i \gt 0, i=1,\dots,k; a \gt 0. \quad (2) }[/math]

The marginal means and variances are given by

[math]\displaystyle{ E[X_i] = \frac{a \theta_i}{a-1}, \text{ for } a \gt 1, \text{ and } Var(X_i) = \frac{a \theta_i^2}{(a-1)^2 (a-2)}, \text{ for } a \gt 2. }[/math]

If a > 2 the covariances and correlations are positive with

[math]\displaystyle{ \operatorname{cov}(X_i, X_j) = \frac{\theta_i \theta_j}{(a-1)^2(a-2)}, \qquad \operatorname{cor}(X_i, X_j) = \frac{1}{a}, \qquad i \neq j. }[/math]

Multivariate Pareto distribution of the second kind

Arnold suggests representing the multivariate Pareto Type I complementary CDF by

[math]\displaystyle{ \overline{F}(x_1, \dots, x_k) = \left(1 + \sum_{i=1}^k \frac{x_i-\theta_i}{\theta_i} \right)^{-a}, \qquad x_i \gt \theta_i\gt 0, \quad i=1,\dots, k. }[/math]

If the location and scale parameter are allowed to differ, the complementary CDF is

[math]\displaystyle{ \overline{F}(x_1,\dots,x_k) = \left(1 + \sum_{i=1}^k \frac{x_i-\mu_i}{\sigma_i} \right)^{-a}, \qquad x_i \gt \mu_i, \quad i=1,\dots,k, \qquad (3) }[/math]

which has marginal distributions of the same type (3) and Pareto Type II univariate marginal distributions. This distribution is called a multivariate Pareto distribution of type II by Arnold.

For a > 1, the marginal means are

[math]\displaystyle{ E[X_i] = \mu_i + \frac{\sigma_i}{a-1}, \qquad i=1,\dots,k, }[/math]

while for a > 2, the variances, covariances, and correlations are the same as for multivariate Pareto of the first kind.

Multivariate Pareto distribution of the fourth kind

A random vector X has a k-dimensional multivariate Pareto distribution of the Fourth Kind if its joint survival function is

[math]\displaystyle{ \overline{F}(x_1,\dots,x_k) = \left( 1 + \sum_{i=1}^k \left(\frac{x_i-\mu_i}{\sigma_i}\right)^{1/\gamma_i}\right)^{-a}, \qquad x_i \gt \mu_i, \sigma_i \gt 0, i=1,\dots,k; a \gt 0. \qquad (4) }[/math]

The k₁-dimensional marginal distributions (k₁<k) are of the same type as (4), and the one-dimensional marginal distributions are Pareto Type IV.

Multivariate Feller–Pareto distribution

A random vector X has a k-dimensional Feller–Pareto distribution if

[math]\displaystyle{ X_i = \mu_i + (W_i / Z)^{\gamma_i}, \qquad i=1,\dots,k, \qquad (5) }[/math]

where

[math]\displaystyle{ W_i \sim \Gamma(\beta_i, 1), \quad i=1,\dots,k, \qquad Z \sim \Gamma(\alpha, 1), }[/math]

are independent gamma variables. The marginal distributions and conditional distributions are of the same type (5); that is, they are multivariate Feller–Pareto distributions. The one–dimensional marginal distributions are of Feller−Pareto type.