Modelling of count data using nonparametric mixtures

Abstract

Nonparametric modelling of count data is partly motivated by the fact that using parametric count models not only runs the risk of model misspecification but also is rather restrictive in terms of local approximation. Accordingly, we present a framework of using nonparametric mixtures for flexible modelling of count data. We consider the use of the least squares function in nonparametric mixture modelling and provide two algorithms for least squares fitting of nonparametric mixtures. Two illustrations of the framework are given, each with a particular nonparametric mixture. One illustration is the use of the nonparametric Poisson mixture for general modelling purposes. The other illustration is concerned with modelling of count data from some decreasing distribution, in which the Poisson mixture distribution is less appropriate, for its fitted distribution might not be a decreasing distribution. We define a mixture distribution called the discrete decreasing beta mixture distribution that always has fitted probabilities conforming with the assumption of decreasing probabilities. Through numerical studies, we demonstrate the performance of nonparametric mixtures as modelling tools.