Constrained nonparametric maximum‐likelihood estimation for mixture models

Abstract

Abstract A nonparametric mixture model specifies that observations arise from a mixture distribution, ∫ f (x, θ) dG (θ), where the mixing distribution G is completely unspecified. A number of algorithms have been developed to obtain unconstrained maximum‐likelihood estimates of G , but none of these algorithms lead to estimates when functional constraints are present. In many cases, there is a natural interest in functional ϕ( G ), such as the mean and variance, of the mixing distribution, and profile likelihoods and confidence intervals for ϕ( G ) are desired. In this paper we develop a penalized generalization of the ISDM algorithm of Kalbfleisch and Lesperance (1992) that can be used to solve the problem of constrained estimation. We also discuss its use in various different applications. Convergence results and numerical examples are given for the generalized ISDM algorithm, and asymptotic results are developed for the likelihood‐ratio test statistics in the multinomial case.