Exponential Series Estimator of Multivariate Densities

Abstract

We present an Exponential Series Estimator (ESE) for multivariate densities. The ESE has an appealing information-theoretic interpretation and ensures positive density estimates. We show that if the logarithm of a density p(x) defined on a bounded support has r continuously differentiable derivatives, the ESE converges to p in the sense of Kullback-Leibler Information Criterion at the optimal minimax rate. The same rate is achieved for the integrated square error. We also derive the almost sure uniform convergence rate. We then establish the asymptotic normality of ESE. We undertake two sets of Monte Carlo experiments. The first experiment examines the performance using bivariate normal mixtures as investigated in Wand and Jones (1993). The second experiment estimates a copula density function. The results demonstrate that the ESE is an efficacious multivariate density estimator, especially for small sample sizes and copula estimation. An empirical application on the joint distribution of stock returns is presented.