On approximations via convolution-defined mixture models

An often-cited fact regarding mixing distributions is that their densities can approximate the densities of any unknown distribution to arbitrary degrees of accuracy provided that the mixing distribution is sufficiently complex. This fact is often not made concrete. We investigate theorems that provide approximation bounds for mixing distributions. Novel connections are drawn between the approximation bounds of mixing distributions and estimation bounds for the maximum likelihood estimator of finite mixtures of location-scale distributions. New approximation and estimation bounds are obtained in the context of finite mixtures of truncated location-scale distributions.