multivariate Pareto distribution

Mardia (1962){{cite journal|author=Mardia, K. V.|title=Multivariate Pareto distributions|journal=Annals of Mathematical Statistics|year=1962|volume=33|issue=3|pages=1008–1015|doi=10.1214/aoms/1177704468|doi-access=free}} defined a bivariate distribution with cumulative distribution function (CDF) given by

:$$

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

: $f(x\_1,\; x\_2)\; =\; (a+1)a(\backslash theta\_1\; \backslash theta\_2)^\{a+1\}(\backslash theta\_2x\_1\; +\; \backslash theta\_1x\_2\; -\; \backslash theta\_1\; \backslash theta\_2)^\{-(a+2)\},$

\qquad x_i \geq \theta_i>0, i=1,2; a>0.

The marginal distributions are Pareto Type 1 with density functions

: $f(x\_i)=a\backslash theta\_i^a\; x\_i^\{-(a+1)\},\; \backslash qquad\; x\_i\; \backslash geq\; \backslash theta\_i>0,\; i=1,2.$

The means and variances of the marginal distributions are

:$E[X\_i]\; =\; \backslash frac\{a\; \backslash theta\_i\}\{a-1\},\; a>1;\; \backslash quad\; Var(X\_i)=\backslash frac\{a\backslash theta\_i^2\}\{(a-1)^2(a-2)\},\; a>2;\; \backslash quad\; i=1,2,$

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

:$\backslash operatorname\{cov\}(X\_1,\; X\_2)\; =\; \backslash frac\{\backslash theta\_1\; \backslash theta\_2\}\{(a-1)^2\; (a-2)\},\; \backslash text\{\; and\; \}$

\operatorname{cor}(X_1, X_2) = \frac{1}{a}.

Arnold{{cite book|author=Barry C. Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn=0-89974-012-X}} Chapter 6. suggests representing the bivariate Pareto Type I complementary CDF by

:$\backslash overline\{F\}(x\_1,x\_2)\; =\; \backslash left(1\; +\; \backslash sum\_\{i=1\}^2\; \backslash frac\{x\_i-\backslash theta\_i\}\{\backslash theta\_i\}\; \backslash right)^\{-a\},\; \backslash qquad\; x\_i\; >\; \backslash theta\_i,\; i=1,2.$

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

:$\backslash overline\{F\}(x\_1,x\_2)\; =\; \backslash left(1\; +\; \backslash sum\_\{i=1\}^2\; \backslash frac\{x\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\; \backslash right)^\{-a\},\; \backslash qquad\; x\_i\; >\; \backslash mu\_i,\; i=1,2,$

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

For a > 1, the marginal means are

:$$

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

:$f(x\_1,\backslash dots,x\_k)\; =\; a(a+1)\backslash cdots(a+k-1)\; \backslash left(\backslash prod\_\{i=1\}^k\; \backslash theta\_i\; \backslash right)^\{-1\}$

\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

:$$

\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

:$$

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

:$$

\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.

Arnold suggests representing the multivariate Pareto Type I complementary CDF by

:$\backslash overline\{F\}(x\_1,\; \backslash dots,\; x\_k)\; =\; \backslash left(1\; +\; \backslash sum\_\{i=1\}^k\; \backslash frac\{x\_i-\backslash theta\_i\}\{\backslash theta\_i\}\; \backslash right)^\{-a\},\; \backslash qquad\; x\_i\; >\; \backslash theta\_i>0,\; \backslash quad\; i=1,\backslash dots,\; k.$

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

:$\backslash overline\{F\}(x\_1,\backslash dots,x\_k)\; =\; \backslash left(1\; +\; \backslash sum\_\{i=1\}^k\; \backslash frac\{x\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\; \backslash right)^\{-a\},\; \backslash qquad\; x\_i\; >\; \backslash mu\_i,\; \backslash quad\; i=1,\backslash dots,k,\; \backslash qquad\; (3)$

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.

For a > 1, the marginal means are

:$$

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 Kind if its joint survival function is

:$\backslash overline\{F\}(x\_1,\backslash dots,x\_k)\; =\; \backslash left(\; 1\; +\; \backslash sum\_\{i=1\}^k\; \backslash left(\backslash frac\{x\_i-\backslash mu\_i\}\{\backslash sigma\_i\}\backslash right)^\{1/\backslash gamma\_i\}\backslash right)^\{-a\},\; \backslash qquad$

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

:$X\_i\; =\; \backslash mu\_i\; +\; (W\_i\; /\; Z)^\{\backslash gamma\_i\},\; \backslash qquad\; i=1,\backslash dots,k,\; \backslash qquad\; (5)$

where

:$W\_i\; \backslash sim\; \backslash Gamma(\backslash beta\_i,\; 1),\; \backslash quad\; i=1,\backslash dots,k,\; \backslash qquad\; Z\; \backslash sim\; \backslash Gamma(\backslash alpha,\; 1),$

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.

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{{ProbDistributions|multivariate}}

{{DEFAULTSORT:Multivariate Pareto distribution}}

Category:Multivariate continuous distributions