Multivariate Pareto distribution

In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution.¹

There are several different types of univariate Pareto distributions including Pareto Types I−IV and Feller−Pareto.² Multivariate Pareto distributions have been defined for many of these types.

Mardia (1962)³ defined a bivariate distribution with cumulative distribution function (CDF) given by

${\displaystyle F(x_{1},x_{2})=1-\sum _{i=1}^{2}\left({\frac {x_{i}}{\theta _{i}}}\right)^{-a}+\left(\sum _{i=1}^{2}{\frac {x_{i}}{\theta _{i}}}-1\right)^{-a},\qquad x_{i}>\theta _{i}>0,i=1,2;a>0,}$

and joint density function

${\displaystyle f(x_{1},x_{2})=(a+1)a(\theta _{1}\theta _{2})^{a+1}(\theta _{2}x_{1}+\theta _{1}x_{2}-\theta _{1}\theta _{2})^{-(a+2)},\qquad x_{i}\geq \theta _{i}>0,i=1,2;a>0.}$

The marginal distributions are Pareto Type 1 with density functions

${\displaystyle f(x_{i})=a\theta _{i}^{a}x_{i}^{-(a+1)},\qquad x_{i}\geq \theta _{i}>0,i=1,2.}$

The means and variances of the marginal distributions are

${\displaystyle E[X_{i}]={\frac {a\theta _{i}}{a-1}},a>1;\quad Var(X_{i})={\frac {a\theta _{i}^{2}}{(a-1)^{2}(a-2)}},a>2;\quad i=1,2,}$

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

${\displaystyle \operatorname {cov} (X_{1},X_{2})={\frac {\theta _{1}\theta _{2}}{(a-1)^{2}(a-2)}},{\text{ and }}\operatorname {cor} (X_{1},X_{2})={\frac {1}{a}}.}$

Arnold⁴ suggests representing the bivariate Pareto Type I complementary CDF by

${\displaystyle {\overline {F}}(x_{1},x_{2})=\left(1+\sum _{i=1}^{2}{\frac {x_{i}-\theta _{i}}{\theta _{i}}}\right)^{-a},\qquad x_{i}>\theta _{i},i=1,2.}$

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

${\displaystyle {\overline {F}}(x_{1},x_{2})=\left(1+\sum _{i=1}^{2}{\frac {x_{i}-\mu _{i}}{\sigma _{i}}}\right)^{-a},\qquad x_{i}>\mu _{i},i=1,2,}$

which has Pareto Type II univariate marginal distributions. This distribution is called a multivariate Pareto distribution of type II by Arnold.⁴ (This definition is not equivalent to Mardia's bivariate Pareto distribution of the second kind.)³

For a > 1, the marginal means are

${\displaystyle E[X_{i}]=\mu _{i}+{\frac {\sigma _{i}}{a-1}},\qquad i=1,2,}$

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

Mardia's³ Multivariate Pareto distribution of the First Kind has the joint probability density function given by

${\displaystyle f(x_{1},\dots ,x_{k})=a(a+1)\cdots (a+k-1)\left(\prod _{i=1}^{k}\theta _{i}\right)^{-1}\left(\sum _{i=1}^{k}{\frac {x_{i}}{\theta _{i}}}-k+1\right)^{-(a+k)},\qquad x_{i}>\theta _{i}>0,a>0,\qquad (1)}$

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

${\displaystyle {\overline {F}}(x_{1},\dots ,x_{k})=\left(\sum _{i=1}^{k}{\frac {x_{i}}{\theta _{i}}}-k+1\right)^{-a},\qquad x_{i}>\theta _{i}>0,i=1,\dots ,k;a>0.\quad (2)}$

The marginal means and variances are given by

${\displaystyle E[X_{i}]={\frac {a\theta _{i}}{a-1}},{\text{ for }}a>1,{\text{ and }}Var(X_{i})={\frac {a\theta _{i}^{2}}{(a-1)^{2}(a-2)}},{\text{ for }}a>2.}$

If a > 2 the covariances and correlations are positive with

${\displaystyle \operatorname {cov} (X_{i},X_{j})={\frac {\theta _{i}\theta _{j}}{(a-1)^{2}(a-2)}},\qquad \operatorname {cor} (X_{i},X_{j})={\frac {1}{a}},\qquad i\neq j.}$

Arnold⁴ suggests representing the multivariate Pareto Type I complementary CDF by

${\displaystyle {\overline {F}}(x_{1},\dots ,x_{k})=\left(1+\sum _{i=1}^{k}{\frac {x_{i}-\theta _{i}}{\theta _{i}}}\right)^{-a},\qquad x_{i}>\theta _{i}>0,\quad i=1,\dots ,k.}$

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

${\displaystyle {\overline {F}}(x_{1},\dots ,x_{k})=\left(1+\sum _{i=1}^{k}{\frac {x_{i}-\mu _{i}}{\sigma _{i}}}\right)^{-a},\qquad x_{i}>\mu _{i},\quad i=1,\dots ,k,\qquad (3)}$

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.⁴

For a > 1, the marginal means are

${\displaystyle E[X_{i}]=\mu _{i}+{\frac {\sigma _{i}}{a-1}},\qquad i=1,\dots ,k,}$

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

A random vector X has a k-dimensional multivariate Pareto distribution of the Fourth Kind⁴ if its joint survival function is

${\displaystyle {\overline {F}}(x_{1},\dots ,x_{k})=\left(1+\sum _{i=1}^{k}\left({\frac {x_{i}-\mu _{i}}{\sigma _{i}}}\right)^{1/\gamma _{i}}\right)^{-a},\qquad x_{i}>\mu _{i},\sigma _{i}>0,i=1,\dots ,k;a>0.\qquad (4)}$

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

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

${\displaystyle X_{i}=\mu _{i}+(W_{i}/Z)^{\gamma _{i}},\qquad i=1,\dots ,k,\qquad (5)}$

where

${\displaystyle W_{i}\sim \Gamma (\beta _{i},1),\quad i=1,\dots ,k,\qquad Z\sim \Gamma (\alpha ,1),}$

are independent gamma variables.⁴ 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.