THC Science

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.

Bivariate Pareto distributions[edit]

Bivariate Pareto distribution of the first kind[edit]

Mardia (1962)^[3] 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(\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

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

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

\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}}.

Bivariate Pareto distribution of the second kind[edit]

Arnold^[4] suggests representing the bivariate Pareto Type I complementary CDF by

{\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

{\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

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.

Multivariate Pareto distributions[edit]

Multivariate Pareto distribution of the first kind[edit]

Mardia's^[3] Multivariate Pareto distribution of the First Kind has the joint probability density function given by

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

{\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.

Multivariate Pareto distribution of the second kind[edit]

Arnold^[4] suggests representing the multivariate Pareto Type I complementary CDF by

{\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

{\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

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.

Multivariate Pareto distribution of the fourth kind[edit]

A random vector X has a k-dimensional multivariate Pareto distribution of the Fourth Kind^[4] if its joint survival function is

{\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.

Multivariate Feller–Pareto distribution[edit]

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

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

where

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.

References[edit]

^ S. Kotz; N. Balakrishnan; N. L. Johnson (2000). "52". Continuous Multivariate Distributions. Vol. 1 (second ed.). ISBN 0-471-18387-3.
^ Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 3.
^ ^a ^b ^c Mardia, K. V. (1962). "Multivariate Pareto distributions". Annals of Mathematical Statistics. 33 (3): 1008–1015. doi:10.1214/aoms/1177704468.
^ ^a ^b ^c ^d ^e ^f Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6.

[CMD1-1] S. Kotz; N. Balakrishnan; N. L. Johnson (2000). "52". Continuous Multivariate Distributions. Vol. 1 (second ed.). ISBN 0-471-18387-3.

[arnold3-2] Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 3.

[Mardia62-3] Mardia, K. V. (1962). "Multivariate Pareto distributions". Annals of Mathematical Statistics. 33 (3): 1008–1015. doi:10.1214/aoms/1177704468.

[arnold6-4] ^ ^a ^b ^c ^d ^e ^f Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6.

[1]

[2]

[3]

[4]