Exponential Series Estimator of multivariate densities

We present an Exponential Series Estimator (ESE) of multivariate densities, which has an appealing information-theoretic interpretation. For a d dimensional random variable x with density p 0 , the ESE takes the form p θ ( x ) = exp ( ∑ i 1 = 0 m 1 ⋯ ∑ i d = 0 m d θ i ϕ i ( x ) ) , where ϕ i are some real-valued, linearly independent functions defined on the support of p 0 . We derive the convergence rate of the ESE in terms of the Kullback–Leibler Information Criterion, the integrated squared error and some other metrics. We also derive its almost sure uniform convergence rate. We then establish the asymptotic normality of p θ ˆ . We undertake two sets of Monte Carlo experiments. The first experiment examines the ESE performance using mixtures of multivariate normal densities. The second estimates copula density functions. The results demonstrate the efficacy of the ESE. An empirical application on the joint distributions of stock returns is presented.