Estimation of a Two-component Mixture Model with Applications to Multiple Testing

TL;DR

This paper introduces nonparametric methods for estimating the mixing proportion and unknown distribution in a two-component mixture model with one known component, providing theoretical guarantees and practical tools for multiple testing applications.

Contribution

It develops consistent, rate-optimal estimators for the mixing proportion and unknown distribution, along with distribution-free confidence bounds, applicable to multiple testing scenarios.

Findings

Estimators are consistent and have known convergence rates.

Developed distribution-free confidence bounds for the mixing proportion.

Validated methods through simulations and real data analysis.

Abstract

We consider a two-component mixture model with one known component. We develop methods for estimating the mixing proportion and the unknown distribution nonparametrically, given i.i.d.~data from the mixture model, using ideas from shape restricted function estimation. We establish the consistency of our estimators. We find the rate of convergence and asymptotic limit of the estimator for the mixing proportion. Completely automated distribution-free honest finite sample lower confidence bounds are developed for the mixing proportion. Connection to the problem of multiple testing is discussed. The identifiability of the model, and the estimation of the density of the unknown distribution are also addressed. We compare the proposed estimators, which are easily implementable, with some of the existing procedures through simulation studies and analyse two data sets, one arising from an application in astronomy and the other from a microarray experiment.