An exponential partial prior for improving nonparametric maximum likelihood estimation in mixture models

Given observations originating from a mixture distribution f[x;Q(λ)]f[x;Q(λ)] where the kernel ff is known and the mixing distribution QQ is unknown, we consider estimating a functional θ(Q)θ(Q) of QQ. A natural estimator of such a functional can be obtained by substituting QQ with its nonparametric maximum likelihood estimator (NPMLE), denoted here as Qˆ. We demonstrate however, that the plug-in estimator θ(Qˆ) can be unstable or substantially biased due to large variability of Qˆ or structural properties of the parameter space of λλ. In this paper we propose using a partial prior   for QQ to improve the estimation in motivating examples. In particular we propose an empirical Bayes estimation method based on an exponential prior, and show its effectiveness in improving estimation in motivating examples of binomial mixture.