Pseudo maximum likelihood estimation in elliptical theory: Effects of misspecification

Abstract

Recently, robust extensions of normal theory statistics have been proposed to permit modeling under a wider class of distributions (e.g., Taylor, 1992). Let X be a p × 1 random vector, μ a p × 1 location parameter, and V a p × p scatter matrix. Kano et al. (1993) studied inference in the elliptical class of distributions and gave a criterion for the choice of a particular family within this class to best describe the data at hand when the latter exhibit serious departure from normality. In this paper, we investigate the criterion for a simple but general set-up, namely, when the operating distribution is multivariate t with ν degrees of freedom and the model is also a multivariate t-distribution with α degrees of freedom. We compute the exact inefficiency of the estimators of μ and V based on that model and compare it to the one based on the mutivariate normal model. Our results provide evidence for the choice of ν = 4 proposed by Lange (1989). In addition, we give numerical results showing that for fixed ν, the inflation of the variance of the pseudo maximum likelihood estimator of the scatter matrix, as a function of the hypothesized degrees of freedom α, is increasing in its domain.