  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.
