## 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,

and

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

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

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

**H**owever, 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?
*

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 [...]

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

with and $k\times k$ symmetric positive definite matrix .

For this lognormal we have

and it follows that .

So, we can ask the converse question: given and symmetric positive definite matrix , satisfying , if we let

we will have a lognormal vector with the prescribed means and covariances.

Regards,

Paulo.

P.S. The constraint on is equivalent to .

April 5, 2012 at 9:59 pm

Terrific! You should also post the reply on StackOverflow, this answers my question!!!

April 6, 2012 at 12:56 am

Ok, done!

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.

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.

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.