On approximations via convolution-defined mixture models

TL;DR

This paper reviews theorems on how convolution-defined mixture models can approximate any distribution's density function with arbitrary accuracy, linking approximation bounds to estimation bounds for finite mixture models.

Contribution

It provides a detailed review of approximation bounds for mixing distributions and connects these bounds to estimation bounds for maximum likelihood estimators in finite mixtures.

Findings

Convolution-defined mixture models can approximate any distribution density arbitrarily well.

Approximation bounds are connected to estimation bounds for maximum likelihood estimators.

Theoretical insights bridge approximation capabilities and estimation performance in mixture models.

Abstract

An often-cited fact regarding mixing or mixture distributions is that their density functions are able to approximate the density function of any unknown distribution to arbitrary degrees of accuracy, provided that the mixing or mixture distribution is sufficiently complex. This fact is often not made concrete. We investigate and review theorems that provide approximation bounds for mixing distributions. Connections between the approximation bounds of mixing distributions and estimation bounds for the maximum likelihood estimator of finite mixtures of location- scale distributions are reviewed.