UNIFORM CONSISTENCY IN NONPARAMETRIC MIXTURE MODELS

• 成果类型: Article
• 署名作者: Aragam, Bryon; Yang, Ruiyi
• 署名单位: University of Chicago; Princeton University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2255
• 发表日期: 2023
• 页码: 362-390
• 关键词: maximum-likelihood; semiparametric mixtures; hierarchical mixtures; parameter-estimation; DENSITY-ESTIMATION; convergence-rates; gaussian mixtures; regression-models; em algorithm; of-experts

摘要

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 point -wise 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 re-gression 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) nonparamet-ric mixtures.