Convergence of Estimative Density to Information Projection for Misspecified Normal Distribution Model

Abstract

This paper investigates the convergence of an estimative multivariate normal density when the true distribution is a misspecified multivariate t-distribution. The statistical model is the family of k-dimensional normal distributions (N_k(\mu,\Sigma)), whereas the observations are assumed to follow (t_k(0,I_k,\nu)), with (\nu>6). The information projection of the true distribution onto the normal model is first identified as the normal distribution with mean zero and covariance matrix (\nu/(\nu-2)I_k). The main objective is to evaluate the expected Kullback-Leibler divergence between this information projection and the normal density obtained by substituting the maximum likelihood estimator into the model. Using a general asymptotic expansion for estimative densities, the paper derives explicit first- and second-order terms of the risk as functions of the sample size (n), the dimension (k), and the degrees of freedom (\nu). To obtain the second-order term, the paper calculates the required moments, information matrices, and higher-order cumulants under both the multivariate normal and multivariate t-distributions. In particular, the complicated third- and fourth-order cumulants involving quadratic sufficient statistics are classified according to their index patterns, and their values and multiplicities are systematically derived.