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

Trungtin Nguyen (1) , Faicel Chamroukhi (2, 1) , Hien D Nguyen (2) , Geoffrey J Mclachlan (3)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 La Trobe University [Melbourne]
3 UQ [All campuses : Brisbane, Dutton Park Gatton, Herston, St Lucia and other locations] - The University of Queensland

The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics. We 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 $p\ge1$, location-scale mixtures of an essentially bounded PDF can approximate any PDF in $\mathcal{L}_{p}$, in the $\mathcal{L}_{p}$ norm.

Domaines: Statistiques [math.ST]

Trungtin Nguyen, Faicel Chamroukhi, Hien D Nguyen, Geoffrey J Mclachlan. Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces. Communications in Statistics - Theory and Methods, 2022, pp. 1-12. ⟨10.1080/03610926.2021.2002360⟩. ⟨hal-02957876⟩