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

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

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

the [corrected, thanks to David Epstein!] solution is (hopefully) given by the system

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

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

$\sigma^2_1\mu_2-\sigma_{12}\mu_1 >0$

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?

This entry was posted on April 4, 2012 at 12:12 am and is filed under R, Statistics, University life. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

7 Responses to “What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? (bis)”

What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? (solved) « Xi'an's Og Says:
April 8, 2012 at 12:13 am

[…] Universidade de São Paulo, Brazil) has posted an answer to my earlier question both as a comment on the ‘Og and as a solution on StackOverflow (with a much more readable LaTeX output). His solution is based […]

Reply
Paulo. Says:
April 5, 2012 at 1:01 am

Dear Xi’an,

I’m not sure if the following helps with your question. Suppose that we have a multivariate normal random vector
$(\log X_1,\dots,\log X_k) \sim N(\mu,\Sigma) \, ,$
with $\mu\in\mathbb{R}^k$ and $k\times k$ symmetric positive definite matrix $\Sigma=(\sigma_{ij})$ .

For this lognormal $(X_1,\dots,X_k)$ we have
$m_i := E[X_i] = e^{\mu_i + \sigma_{ii}/2} \, , \quad i=1,\dots,k\, ,$
$c_{ij} := Cov[X_i,X_j] = m_i \,m_j \,(e^{\sigma_{ij}} - 1) \, , \quad i,j=1,\dots,k\, ,$
and it follows that $c_{ij}\ge -m_im_j$ .

So, we can ask the converse question: given $m=(m_1,\dots,m_k)\in\mathbb{R}^k_+$ and $k\times k$ symmetric positive definite matrix $C=(c_{ij})$ , satisfying $c_{ij}>-m_im_j$ , if we let
$\mu_i = \log m_i - \frac{1}{2} \log\left(\frac{c_{ii}}{m_i^2} + 1 \right) \, , \quad i=1,\dots,k \, ,$
$\sigma_{ij} = \log\left(\frac{c_{ij}}{m_i m_j} + 1 \right) \, , \quad i,j=1,\dots,k \, ,$
we will have a lognormal vector with the prescribed means and covariances.

Regards,

Paulo.

P.S. The constraint on $C$ is equivalent to $\mathbb{E}[X_i X_j]\ge 0$ .

Reply
- xi'an Says:
  April 5, 2012 at 9:59 pm
  
  Terrific! You should also post the reply on StackOverflow, this answers my question!!!
  
  Reply
  - Paulo. Says:
    April 6, 2012 at 12:56 am
    
    Ok, done!
deinst Says:
April 4, 2012 at 3:04 am

I have opened a question on math.se http://math.stackexchange.com/questions/127813/what-are-the-restrictions-on-the-covariance-matrix-of-a-nonnegative-multivariate

I suspect that someone there may be able to answer your initial question as well.

Reply
- xi'an Says:
  April 4, 2012 at 10:01 am
  
  Thank you: I was planning to post this question myself, and would have preferred to do so, but this is not a major problem! We will see if this attracts more answers than my original question.
  
  Reply
  - Deinst Says:
    April 4, 2012 at 6:09 pm
    
    I apologize.
    
    One comment: Your R code does not seem to restrict the covariance of the two variables to be greater than -mu[1]mu[2]. This follows from Cov(X,Y) = E(XY)-E(X)E(Y) and the fact that XY is always nonnegative.