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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 − ∑ i = 1 2 ( x i θ i ) − a + ( ∑ i = 1 2 x i θ i − 1 ) − a , x i > θ i > 0 , i = 1 , 2 ; a > 0 , {\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

f ( x 1 , x 2 ) = ( a + 1 ) a ( θ 1 θ 2 ) a + 1 ( θ 2 x 1 + θ 1 x 2 − θ 1 θ 2 ) − ( a + 2 ) , x i ≥ θ i > 0 , i = 1 , 2 ; a > 0. {\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

f ( x i ) = a θ i a x i − ( a + 1 ) , x i ≥ θ i > 0 , i = 1 , 2. {\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

E [ X i ] = a θ i a − 1 , a > 1 ; V a r ( X i ) = a θ i 2 ( a − 1 ) 2 ( a − 2 ) , a > 2 ; i = 1 , 2 , {\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

cov ⁡ ( X 1 , X 2 ) = θ 1 θ 2 ( a − 1 ) 2 ( a − 2 ) ,  and  cor ⁡ ( X 1 , X 2 ) = 1 a . {\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}}.}

Bivariate Pareto distribution of the second kind

Arnold⁴ suggests representing the bivariate Pareto Type I complementary CDF by

F ¯ ( x 1 , x 2 ) = ( 1 + ∑ i = 1 2 x i − θ i θ i ) − a , x i > θ i , i = 1 , 2. {\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

F ¯ ( x 1 , x 2 ) = ( 1 + ∑ i = 1 2 x i − μ i σ i ) − a , x i > μ i , i = 1 , 2 , {\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.⁵ (This definition is not equivalent to Mardia's bivariate Pareto distribution of the second kind.)⁶

For a > 1, the marginal means are

E [ X i ] = μ i + σ i a − 1 , i = 1 , 2 , {\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.

Multivariate Pareto distributions

Multivariate Pareto distribution of the first kind

Mardia's⁷ 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 ) ( ∏ i = 1 k θ i ) − 1 ( ∑ i = 1 k x i θ i − k + 1 ) − ( a + k ) , x i > θ i > 0 , a > 0 , ( 1 ) {\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

F ¯ ( x 1 , … , x k ) = ( ∑ i = 1 k x i θ i − k + 1 ) − a , x i > θ i > 0 , i = 1 , … , k ; a > 0. ( 2 ) {\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

E [ X i ] = a θ i a − 1 ,  for  a > 1 ,  and  V a r ( X i ) = a θ i 2 ( a − 1 ) 2 ( a − 2 ) ,  for  a > 2. {\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

cov ⁡ ( X i , X j ) = θ i θ j ( a − 1 ) 2 ( a − 2 ) , cor ⁡ ( X i , X j ) = 1 a , i ≠ j . {\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.}

Multivariate Pareto distribution of the second kind

Arnold⁸ suggests representing the multivariate Pareto Type I complementary CDF by

F ¯ ( x 1 , … , x k ) = ( 1 + ∑ i = 1 k x i − θ i θ i ) − a , x i > θ i > 0 , i = 1 , … , k . {\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

F ¯ ( x 1 , … , x k ) = ( 1 + ∑ i = 1 k x i − μ i σ i ) − a , x i > μ i , i = 1 , … , k , ( 3 ) {\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.⁹

For a > 1, the marginal means are

E [ X i ] = μ i + σ i a − 1 , i = 1 , … , k , {\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.

Multivariate Pareto distribution of the fourth kind

A random vector X has a k-dimensional multivariate Pareto distribution of the Fourth Kind¹⁰ if its joint survival function is

F ¯ ( x 1 , … , x k ) = ( 1 + ∑ i = 1 k ( x i − μ i σ i ) 1 / γ i ) − a , x i > μ i , σ i > 0 , i = 1 , … , k ; a > 0. ( 4 ) {\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.

Multivariate Feller–Pareto distribution

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

X i = μ i + ( W i / Z ) γ i , i = 1 , … , k , ( 5 ) {\displaystyle X_{i}=\mu _{i}+(W_{i}/Z)^{\gamma _{i}},\qquad i=1,\dots ,k,\qquad (5)}

where

W i ∼ Γ ( β i , 1 ) , i = 1 , … , k , Z ∼ Γ ( α , 1 ) , {\displaystyle W_{i}\sim \Gamma (\beta _{i},1),\quad i=1,\dots ,k,\qquad Z\sim \Gamma (\alpha ,1),}

are independent gamma variables.¹¹ 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

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

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

3. Mardia, K. V. (1962). "Multivariate Pareto distributions". Annals of Mathematical Statistics. 33 (3): 1008–1015. doi:10.1214/aoms/1177704468. https://doi.org/10.1214%2Faoms%2F1177704468 ↩

4. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6. 0-89974-012-X ↩

5. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6. 0-89974-012-X ↩

6. Mardia, K. V. (1962). "Multivariate Pareto distributions". Annals of Mathematical Statistics. 33 (3): 1008–1015. doi:10.1214/aoms/1177704468. https://doi.org/10.1214%2Faoms%2F1177704468 ↩

7. Mardia, K. V. (1962). "Multivariate Pareto distributions". Annals of Mathematical Statistics. 33 (3): 1008–1015. doi:10.1214/aoms/1177704468. https://doi.org/10.1214%2Faoms%2F1177704468 ↩

8. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6. 0-89974-012-X ↩

9. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6. 0-89974-012-X ↩

10. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6. 0-89974-012-X ↩

11. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 0-89974-012-X. Chapter 6. 0-89974-012-X ↩