Posterior predictive distribution: Difference between revisions

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In [[statistics]], a '''multivariate Pareto distribution''' is a multivariate extension of a univariate [[Pareto distribution]].<ref name=CMD1>{{cite book|title=Continuous Multivariate Distributions|volume=1|edition=second|author=S. Kotz, N. Balakrishnan, N. L. Johnson|chapter=52|year=2000|isbn=0-471-18387-3|ref=harv}}</ref>
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There are several different types of univariate Pareto distributions including [[Pareto_distribution#Pareto types I–IV|Pareto Types I−IV]] and [[Pareto_distribution#Feller–Pareto_distribution|Feller−Pareto]].<ref name=arnold3>{{cite book|author=Barry C. Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn=0-89974-012-X}} Chapter 3.</ref> Multivariate Pareto distributions have been defined for many of these types.
 
==Bivariate Pareto distributions==
 
===Bivariate Pareto distribution of the first kind===
 
Mardia (1962)<ref name=Mardia62>{{cite journal|author=Mardia, K. V.|title=Multivariate Pareto distributions|journal=Annals of Mathematical Statistics|volume=33|pages=1008–1015|ref=harv}}</ref> defined a bivariate distribution with cumulative distribution function (CDF) given by
:<math>
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,
</math>
 
and joint density function
 
: <math> f(x_1, x_2) = (a+1)a(\theta_1 \theta_2)^{a+1}(\theta_2x_1 + \theta_1x_2 - \theta_1 \theta_2)^{-(a+2)},
\qquad x_i \geq \theta_i>0, i=1,2; a>0.</math>
The marginal distributions are [[Pareto distribution|Pareto Type 1]] with density functions
 
: <math> f(x_i)=a\theta_i^a x_i^{-(a+1)}, \qquad x_i \geq \theta_i>0, i=1,2.</math>
 
The means and variances of the marginal distributions are
:<math> 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,</math>
and for ''a'' > 2, ''X''<sub>1</sub> and ''X''<sub>2</sub> are positively correlated with
:<math> \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}.
</math>
 
===Bivariate Pareto distribution of the second kind===
 
Arnold<ref name=arnold6>{{cite book|author=Barry C. Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn=0-89974-012-X|ref=harv}} Chapter 6.</ref>  suggests representing the bivariate Pareto Type I complementary CDF by
 
:<math> \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.
</math>
If the location and scale parameter are allowed to differ, the complementary CDF is
 
:<math> \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,
</math>
 
which has Pareto Type II univariate marginal distributions. This distribution is called a '''multivariate Pareto distribution of type II''' by Arnold.<ref name=arnold6/> (This definition is not equivalent to Mardia's bivariate Pareto distribution of the second kind.)<ref name=Mardia62/>
 
For ''a'' > 1, the marginal means are
:<math>
E[X_i] = \mu_i + \frac{\sigma_i}{a-1}, \qquad i=1,2,
</math>
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<ref name=Mardia62/> ''Multivariate Pareto distribution of the First Kind'' has the joint probability density function given by
 
:<math> 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)
</math>
 
The marginal distributions have the same form as (1), and the one-dimensional marginal distributions have a [[Pareto distribution|Pareto Type I distribution]]. The complementary CDF is
:<math>
\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)
</math>
 
The marginal means and variances are given by
:<math>
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.
</math>
If ''a'' > 2 the covariances and correlations are positive with
 
:<math>
\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.
</math>
 
===Multivariate Pareto distribution of the second kind===
 
Arnold<ref name=arnold6/> suggests representing the multivariate Pareto Type I complementary CDF by
 
:<math> \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. </math>
If the location and scale parameter are allowed to differ, the complementary CDF is
 
:<math> \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)
</math>
 
which has marginal distributions of the same type (3) and [[Pareto_distribution#Pareto types I–IV|Pareto Type II]] univariate marginal distributions. This distribution is called a '''multivariate Pareto distribution of type II''' by Arnold.<ref name=arnold6/>
 
For ''a'' > 1, the marginal means are
:<math>
E[X_i] = \mu_i + \frac{\sigma_i}{a-1}, \qquad i=1,\dots,k,
</math>
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'''<ref name=arnold6/> if its joint survival function is
:<math> \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)
</math>
The ''k''<sub>1</sub>-dimensional marginal distributions (''k<sub>1</sub><''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
:<math> X_i = \mu_i + (W_i / Z)^{\gamma_i}, \qquad i=1,\dots,k,  \qquad (5)
</math>
where
:<math> W_i \sim \Gamma(\beta_i, 1), \quad i=1,\dots,k, \qquad Z \sim \Gamma(\alpha, 1),
</math>
are independent gamma variables.<ref name=arnold6/> 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 [[Pareto_distribution#Feller–Pareto_distribution|Feller−Pareto]]  type.
 
==References==
{{Reflist}}
 
{{ProbDistributions|multivariate}}
 
{{DEFAULTSORT:Multivariate Pareto distribution}}
[[Category:Multivariate continuous distributions]]
[[Category:Probability distributions]]

Latest revision as of 13:08, 17 August 2014

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