Uniform Consistency in Nonparametric 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. nonconvolutional) nonparametric mixtures.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 DOI:
 10.48550/arXiv.2108.14003
 arXiv:
 arXiv:2108.14003
 Bibcode:
 2021arXiv210814003A
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Machine Learning
 EPrint:
 To appear in The Annals of Statistics