What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? (bis)

Posted on April 3, 2012 by xi'an in R bloggers | 0 Comments

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Wondering about the question I posted on Friday (on StackExchange, no satisfactory answer so far!), I looked further at the special case of the gamma distribution I suggested at the end. Starting from the moment conditions,

$\dfrac{\alpha_{11}}{\beta_1} = \mu_1\,,\quad \dfrac{\alpha_{11}}{\beta_1^2} = \sigma_1^2\,,$

$\dfrac{\alpha_{21}\mu_1+\alpha_{22}}{\beta_2} = \mu_2\,,\quad \dfrac{\alpha_{21}^2\sigma^2_1}{\beta_2^2}+\dfrac{\alpha_{21}\mu_1+\alpha_{22}}{\beta_2^2} = \sigma^2_2\,,$

and

$\dfrac{\alpha_{21}(\sigma^2_1+\mu_1^2)+\alpha_{22}}{\beta_2} = \sigma_{12}+\mu_1\mu_2$

the solution is (hopefully) given by the system

$\begin{cases} \beta_1 =\mu_1/\sigma_1^2&\\ \alpha_{11}-\mu_1\beta_1 =0&\\ \alpha_{22} = \mu_2\beta_2 - \alpha_{21}\mu_1&\\ \alpha_{21} = \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)}{\sigma^2_1+\mu_1^2- \mu_1}\beta_2\\ \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)^2}{(\sigma^2_1+\mu_1^2- \mu_1)^2} \sigma_1^2 + \dfrac{\mu_2}{\beta_2} = \sigma^2_2&\\ \end{cases}$

The resolution of this system obviously imposes conditions on those moments, like

$\sigma^2_2 - \dfrac{(\sigma_{12} \mu_1\mu_2-\mu_2)^2}{(\sigma^2_1 \mu_1^2- \mu_1)^2} \sigma_1^2 >0$ 0′ title=’\sigma^2_2 – \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)^2}{(\sigma^2_1+\mu_1^2- \mu_1)^2} \sigma_1^2 >0′ class=’latex’ />

So I ran a small R experiment checking when there was no acceptable solution to the system. I started with five moments that satisfied the basic Stieltjes and determinant conditions

# basically anything
mu=runif(2,0,10)
# Jensen inequality
sig=c(mu[1]^2/runif(1),mu[2]^2/runif(1))
# my R code returning the solution if any
sol(mu,c(sig,runif(1,-sqrt(prod(sig)),sqrt(prod(sig)))))

and got a fair share (20%) of rejections, e.g.

> sol(mu,c(sig,runif(1,-sqrt(prod(sig)),sqrt(prod(sig)))))
$solub
[1] FALSE

$alpha
[1]  0.8086944  0.1220291 -0.1491023

$beta
[1] 0.1086459 0.5320866

However, not being sure about the constraints on the five moments I am now left with another question: what are the necessary and sufficient conditions on the five moments of a pair of positive vectors?! Or, more generally, what are the necessary and sufficient conditions on the k-dimensional μ and Σ for them to be first and second moments of a positive k-dimensional vector?

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