Uniform Consistency in Nonparametric Mixture Models
This work addresses theoretical challenges in statistical estimation for nonparametric mixtures, which is incremental but important for advancing consistency results in mixture modeling.
The paper tackles the problem of establishing uniform consistency for nonparametric mixture models and mixed regression models, proving that uniformly consistent estimators can be constructed under general conditions, with results including $L^1$ convergence of regression functions even when components intersect arbitrarily often.
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.