Multivariate Pareto distribution explained

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

Bivariate Pareto distribution of the first kind

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

F(x _{1,} x _{2)} = 1 - \sum ^{2 \left( (x _{i})/(\theta _{i})} _{i =1} \right) ^{-a} + \left(\sum ^{2} _{i =1} (x _{i})/(\theta _{i}) - 1 \right) ^{-a} , 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 ^{a +1} _{2 )} (\theta _{2x} _{1} + \theta _{1x} _{2} - \theta _{1} \theta ^{- (a +2 )} _{2 )} , x _{i} \geq \theta _{i>0,} i =1 , 2 ; a> 0 . The marginal distributions are Pareto Type 1 with density functionsf(x _{i)=a\theta} ^{a} ^{i} x ^{- (a +1 )} _{i} , x _{i} \geq \theta _{i>0,} i =1 , 2 .

The means and variances of the marginal distributions are

E[X _{i]} = (a \theta _{i})/(a -1) , a> 1 ; Var(X _{i) = (a\theta ^{2} _{i})/((a -1 ) ^{2(a-2)})} , a> 2 ; i =1 , 2 , and for a > 2, X₁ and X₂ are positively correlated with\operatorname{cov}(X _{1,} X _{2)} = (\theta _{1} \theta _{2})/((a -1 ) ^{2} (a -2 )) , and \operatorname{cor}(X _{1,} X _{2)} = (1)/(a) .

Bivariate Pareto distribution of the second kind

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

\overline{F}(x _{1,x} _{2)} = \left( 1 + \sum ^{2} _{i =1} (x _{i-\theta} _{i})/(\theta _{i}) \right) ^{-a} , 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 ^{2} _{i =1} (x _{i-\mu} _{i})/(\sigma _{i}) \right) ^{-a} , 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} + (\sigma _{i})/(a -1) , 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^[3] Multivariate Pareto distribution of the First Kind has the joint probability density function given by

f(x _{1,...,x} _{k)} = a(a +1 ) … (a + k -1 ) \left(\prod ^{k} _{i =1} \theta _{i} \right) ^{-1} \left(\sum ^{k} _{i =1} (x _{i})/(\theta _{i}) - k + 1 \right) ^{-(a+k)} , x _{i} > \theta _{i} > 0 , a > 0 , ( 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,...,x} _{k)} = \left(\sum ^{k} _{i =1} (x _{i})/(\theta _{i}) - k +1 \right) ^{-a} , x _{i} > \theta _{i} > 0 , i =1 , ... ,k; a > 0 . ( 2 )

The marginal means and variances are given by

E[X _{i]} = (a \theta _{i})/(a -1) , for a > 1 , and Var(X _{i)} = (a \theta ^{2} _{i})/((a -1 ) ^{2} (a -2 )) , for a > 2 . If a > 2 the covariances and correlations are positive with\operatorname{cov}(X _{i,} X _{j)} = (\theta _{i} \theta _{j})/((a -1 ) ^{2(a-2)}) , \operatorname{cor}(X _{i,} X _{j)} = (1)/(a) , i ≠ j.

Multivariate Pareto distribution of the second kind

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

\overline{F}(x _{1,} ... , x _{k)} = \left( 1 + \sum ^{k} _{i =1} (x _{i-\theta} _{i})/(\theta _{i}) \right) ^{-a} , x _{i} > \theta _{i>0,} i =1 , ... , k. If the location and scale parameter are allowed to differ, the complementary CDF is\overline{F}(x _{1,...,x} _{k)} = \left( 1 + \sum ^{k} _{i =1} (x _{i-\mu} _{i})/(\sigma _{i}) \right) ^{-a} , x _{i} > \mu _{i,} i =1 , ... ,k, ( 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} + (\sigma _{i})/(a -1) , i =1 , ... ,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^[4] if its joint survival function is

\overline{F}(x _{1,...,x} _{k)} = \left( 1 + \sum ^{k} _{i =1} \left( (x _{i-\mu} _{i})/(\sigma _{i}) \right) ^{1 /\gamma _{i}} \right) ^{-a} , x _{i} > \mu _{i,} \sigma _{i} > 0 , i =1 , ... ,k; a > 0 . ( 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

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

X _{i} = \mu _{i} + (W _{i} / Z) ^{\gamma _{i}} , i =1 , ... ,k, ( 5 ) whereW _{i} \sim \Gamma(\beta _{i,} 1 ), i =1 , ... ,k, 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.

Notes and References

1. Book: Continuous Multivariate Distributions. 1. second. S. Kotz . N. Balakrishnan . N. L. Johnson . 52. 2000. 0-471-18387-3.
2. Book: Barry C. Arnold . 1983 . Pareto Distributions . International Co-operative Publishing House . 0-89974-012-X. Chapter 3.
3. Mardia, K. V.. Multivariate Pareto distributions. Annals of Mathematical Statistics. 1962. 33. 3. 1008–1015. 10.1214/aoms/1177704468. free.
4. Book: Barry C. Arnold . 1983 . Pareto Distributions . International Co-operative Publishing House . 0-89974-012-X. Chapter 6.