Uniform Consistency in Nonparametric Mixture Models

TL;DR

This paper establishes uniform consistency results for nonparametric mixture models and mixed regression models with nonparametric functions, developing new technical tools and addressing challenges like intersecting regression components.

Contribution

It provides the first uniform consistency estimators for nonparametric mixture and mixed regression models under broad conditions, including cases with intersecting regression functions.

Findings

Constructed uniformly consistent estimators for nonparametric mixtures.

Proved $L^1$ convergence of regression functions with intersecting components.

Extended results to general non-convolutional mixture models.

Abstract

We study uniform consistency in nonparametric mixture models as well as closely related mixture of regression (also known as mixed regression) models, where the regression functions are allowed to be nonparametric and the error distributions are assumed to be convolutions of a Gaussian density. We construct uniformly consistent estimators under general conditions while simultaneously highlighting several pain points in extending existing pointwise consistency results to uniform results. The resulting analysis turns out to be nontrivial, and several novel technical tools are developed along the way. In the case of mixed regression, we prove $L^1$ convergence of the regression functions while allowing for the component regression functions to intersect arbitrarily often, which presents additional technical challenges. We also consider generalizations to general (i.e. non-convolutional) nonparametric mixtures.