Efficient semiparametric estimation and model selection for multidimensional mixtures

TL;DR

This paper develops an efficient semiparametric estimation method for multidimensional mixture models, demonstrating asymptotic efficiency, Bayesian properties, and a practical partition selection procedure validated through simulations.

Contribution

It introduces a novel approach to estimate mixture weights efficiently and proposes a data-driven partition selection method with theoretical guarantees.

Findings

Asymptotic efficiency of the MLE under slow partition refinement

Bayesian posterior satisfies a semiparametric Bernstein von Mises theorem

Partition selection procedure satisfies an oracle inequality

Abstract

In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein von Mises theorem. We then propose a cross-validation like procedure to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results.