Global Properties of Kernel Estimators for Mixing Densities in Discrete Exponential Family Models

This paper concerns the global performance of modifications of the kernel estimators considered in Zhang (1995) for a mixing density function g based on a sample from f(x) = ∫ f(x|θ)g(θ)dθ under weighted L-loss, 1 ≤ p ≤ ∞, where f(x|θ) is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive upper bounds for the rate of convergence of the kernel estimators and lower bounds for the optimal convergence rate over various smoothness classes of mixing density functions. In particular under mild conditions, it is shown that these estimators achieve the optimal rate of convergence for the negative binomial mixture and are almost optimal for the Poisson mixture. Global estimation of the mixing distribution function under weighted L-loss is also considered.