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.

翻译：我们研究的是非对称混合物模型的统一一致性,以及密切相关的回归(又称混合回归)模型的混合(又称混合回归)模型,在这些模型中,回归功能被允许为非对称函数,误差分布被假定为高斯密度的演化;我们在一般条件下构建一致的估算器,同时强调将现有点一致性结果扩展至统一结果的若干痛苦点;由此得出的分析结果证明是非技术性的,并沿途开发了若干新的技术工具;在混合回归的情况下,我们证明回归功能的趋同值为1美元,同时允许部分回归功能经常被任意地相互分割,这带来了额外的技术挑战;我们还考虑将一般(即非革命)非参数的混合物普遍化为非参数性混合物。