A mixture model-based nonparametric approach to estimating a count distribution

Abstract

The density of a mixture distribution with unknown discrete mixing distribution can be a way of finding a nonparametric estimate of a density. Comparing with a standard parametric approach that runs the risk of model misspecification and a kernel-based nonparametric approach that retains all data points for constructing a density estimate, this mixture model-based nonparametric approach that naturally circumvents these problems is very appealing to density estimation. Owing to these advantages, this approach to estimating a count distribution is considered. Estimation of a count distribution via this means necessarily involves the problem of nonparametric estimation of a discrete mixing distribution. The nonparametric estimation problem is formulated using the family of power divergences, thus offering a class of nonparametric estimators of a discrete mixing distribution. A fast algorithm is presented for computing the class of nonparametric minimum power divergence estimates with which the corresponding class of mixture model-based estimates of a count distribution can be implemented. Only the use of the probability mass function of the Poisson mixture distribution with unknown discrete mixing distribution is illustrated. The simulation results show that the mixture model-based estimators on average outperform the kernel-based estimators for uncontaminated samples of larger sizes and contaminated samples of all sizes considered.