Recovery of Distributions via Moments

Abstract

The problem of recovering a cumulative distribution function (cdf) and corresponding density function from its moments is studied. This problem is a special case of the classical moment problem. The results obtained within the moment problem can be applied in many indirect models, e.g., those based on convolutions, mixtures, multiplicative censoring, and right-censoring, where the moments of unobserved distribution of actual interest can be easily estimated from the transformed moments of the observed distributions. Nonparametric estimation of a quantile function via moments of a target distribution represents another very interesting area where the moment problem arises. In all such models one can apply the present results to recover a function via its moments. In this article some properties of the proposed constructions are derived. The uniform rates of convergence of the approximation of cdf, its density function, quantile and quantile density function are obtained as well.