Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces

Abstract

The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics. We establish and prove the following results: to an arbitrary degree of accuracy, (a) location-scale mixtures of a continuous probability density function (PDF) can approximate any continuous PDF, uniformly, on a compact set; and (b) for any finite $p\ge 1,$ location-scale mixtures of an essentially bounded PDF can approximate any PDF in ${\mathcal{L}}_p,$ in the ${\mathcal{L}}_p$ norm.

Keywords:

• Mixture models
• approximation theory
• uniform approximation
• probability density functions

MATHEMATICS SUBJECT CLASSIFICATION:

• 62E17 (Primary)
• 41A30 (Secondary)

Acknowledgments

The authors would like to very much thank Pr. Eric Ricard for the interesting discussions with him and for his suggestions.

Additional information

Funding

TTN is supported by “Contrat doctoral” from the French Ministry of Higher Education and Research and by the French National Research Agency (ANR) grant SMILES ANR-18-CE40-0014. HDN and GJM are funded by Australian Research Council grant number DP180101192.