Covariance selection and multivariate dependence

Considering the covariance selection problem of multivariate normal distributions, we show that its Fenchel dual formulation is insightful and allows one to calculate direct estimates under decomposable models. We next generalize the covariance selection to multivariate dependence, which includes MTP2 and trends in longitudinal studies as special cases. The iterative proportional scaling algorithm, used for estimation in covariance selection problems, may not lead to the correct solution under such dependence. Addressing this situation, we present a new algorithm for dependence models and show that it converges correctly using tools from Fenchel duality. We discuss the speed of convergence of the new algorithm. When normality does not hold, we show how to estimate the covariance matrix in an empirical entropy approach. The approaches are compared via simulation and it is shown that the estimators developed here compare favorably with existing ones. The methodology is applied on a real data set involving decreasing CD4+ cell numbers from an AIDS study.