Approximation of Distribution Functions: a Constructive Scheme with Application in Bayesian Nonparametrics

Many statistical nonparametric techniques are based on the possibility of approximating a curve of interest (for example, the distribution function generating the data) by basis-functions expansions. In this paper we discuss the problem of constructively approximating a deterministic or random probability distribution function by means of a sequence of distribution functions. Our proposal is based on a general approximation scheme referred to Feller, which we show to have nice probabilistic properties when its basic elements are chosen in the natural exponential family. Some new results for this family are proved, which might be of autonomous interest; exploiting these properties, we can show connections between the proposed scheme and pproximation by mixtures. When the distribution function to be approximated is random, our procedure can be used for prior elicitation in Bayesian nonparametric inference. On one hand, it provides a constructive smoothing of discrete prior processes (e.g. the Dirichlet process), so allowing to elicit a prior more suitable for nonparametric inference with continuous data. Also, it selects distribution functions which have the rather natural form of mixtures. We have thus a general framework which includes continuous, countable and finite mixtures (with an unknown number of components), and various choices of the kernel. Therefore, properties can be studied in a unified setting for a fairly large class of mixture models; in particular, we give a result on weak consistency of the proposed mixture prior