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

Paulo (from the Instituto de Matemática e Estatística, 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 on the observation that the multidimensional log-normal distribution still allows for closed form expressions of both the mean and the variance and that those expressions can further be inverted to impose the pair (μ,Σ)  on the log-normal vector. In addition, he shows that the only constraint on the covariance matrix is that the covariance σ_ij is larger than -μ_iμ_j.. Very neat!

In the meanwhile, I corrected my earlier R code on the gamma model, thanks to David Epstein pointing a mistake in the resolution of the moment equation and I added the constraint on the covariance, already noticed by David in his question. Here is the full code:

sol=function(mu,sigma){
  solub=TRUE
  alpha=rep(0,3)
  beta=rep(0,2)
  beta[1]=mu[1]/sigma[1]
  alpha[1]=mu[1]*beta[1]
  coef=mu[2]*sigma[1]-mu[1]*sigma[3]
  if (coef<0){
   solub=FALSE}else{
    beta[2]=coef/(sigma[1]*sigma[2]-sigma[3]^2)
    alpha[2]=sigma[3]*beta[2]/sigma[1]^2
    alpha[3]=mu[2]*beta[2]-mu[1]*alpha[2]
    if (alpha[3]    }
list(solub=solub,alpha=alpha,beta=beta)
}

mu=runif(2,0,10);sig=c(mu[1]^2/runif(1),mu[2]^2/runif(1));sol(mu,c(sig,runif(1,max(-sqrt(prod(sig)),
-mu[1]*mu[2]),sqrt(prod(sig)))))

and I did not get any FALSE outcome when running this code several times.

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

This is the question I posted this morning on StackOverflow, following an exchange two days ago with a user who could not see why the linear transform of a log-normal vector X,

Y = μ + Σ X

could lead to negative components in Y…. After searching a little while, I could not think of a joint distribution on the positive k-dimensional quadrant where I could specify the covariance matrix in advance. Except for a pedestrian construction of (x₁,x₂) where x₁ would be an arbitrary Gamma variate [with a given variance] and x₂ conditional on x₁ would be a Gamma variate with parameters specified by the covariance matrix. Which does not extend nicely to larger dimensions.

Estimation of covariance matrices

Mathilde Bouriga and Olivier Féron have posted a paper on arXiv centred on the estimation of covariance matrices using inverse-Wishart priors. They introduce hyperpriors on the hyperparameters in the spirit of Daniels and Kass (JASA, 1999) and derive Bayes estimators as well as MCMC procedures. They then run a simulation comparison between the different priors in terms of frequentist risks, concluding in favour of the shrinkage covariance estimators that shrink all components of the empirical covariance matrix. (This paper is part of Mathilde’s thesis, which I co-advise with Jean-Michel Marin.)

More among interesting postings on arXiv, many of them published in Statistical Science:

• Variable Selection for Nonparametric Gaussian Process Priors: Models and Computational Strategies by Terrance Savitsky, Marina Vannucci, Naijun Sha
• Statistical Inference: The Big Picture by Robert E. Kass
• Bayesian Statistical Pragmatism by Andrew Gelman (a discussion of the above)
• A flexible observed factor model with separate dynamics for the factor volatilities and their correlation matrix by Yu-Cheng Ku, Peter Bloomfield, Robert Kohn
• Test martingales, Bayes factors and p-values by Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, Vladimir Vovk
• Pitman-Yor Diffusion Trees by David A. Knowles, Zoubin Ghahramani