title: Multivariate Pareto distribution
text: # 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
  
      $F(x_{1},x_{2})=1-\sum _{i=1}^{2}\left({\frac {x_{i}}{\theta _{i}}}\right)^{-a}+\left(\sum _{i=1}^{2}{\frac {x_{i}}{\theta _{i}}}-1\right)^{-a},\qquad x_{i}>\theta _{i}>0,i=1,2;a>0,$
  
  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, $X_1$ and $X_2$ 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
  
  Arnold 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. (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.
  
  ## 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
  
      $f(x_{1},\dots ,x_{k})=a(a+1)\cdots (a+k-1)\left(\prod _{i=1}^{k}\theta _{i}\right)^{-1}\left(\sum _{i=1}^{k}{\frac {x_{i}}{\theta _{i}}}-k+1\right)^{-(a+k)},\qquad x_{i}>\theta _{i}>0,a>0,\qquad (1)$
  
  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
  
  Arnold 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.
  
  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
  
  A random vector X has a k-dimensional multivariate Pareto distribution of the Fourth Kind 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 $k_1-dimensional$ marginal distributions $(k_1<k)$ 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
  
      $X_{i}=\mu _{i}+(W_{i}/Z)^{\gamma _{i}},\qquad i=1,\dots ,k,\qquad (5)$
  
  where
  
      $W_{i}\sim \Gamma (\beta _{i},1),\quad i=1,\dots ,k,\qquad Z\sim \Gamma (\alpha ,1),$
  
  are independent gamma variables. 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.
  
  