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 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., nonconvolutional) nonparametric mixtures.
B.A. was supported by NSF IIS-1956330, NIH R01GM140467, and the Robert H. Topel Faculty Research Fund at the University of Chicago Booth School of Business.
The authors would like to thank the anonymous reviewers for their constructive comments, which substantially improved the presentation of the paper.
Bryon Aragam. Ruiyi Yang. "Uniform consistency in nonparametric mixture models." Ann. Statist. 51 (1) 362 - 390, February 2023. https://doi.org/10.1214/22-AOS2255