Adaptive density estimation for clustering with Gaussian mixtures

TL;DR

This paper introduces an adaptive density estimation method for Gaussian mixture models that automatically adjusts to the smoothness of the underlying data, improving clustering accuracy.

Contribution

It extends previous work by proving minimax adaptivity of a penalized estimator for a broad class of univariate densities with local smoothness.

Findings

The estimator is minimax adaptive to the density's regularity.

It applies to densities with locally Hölder continuous logarithms.

The method enhances model-based clustering by better density estimation.

Abstract

Gaussian mixture models are widely used to study clustering problems. These model-based clustering methods require an accurate estimation of the unknown data density by Gaussian mixtures. In Maugis and Michel (2009), a penalized maximum likelihood estimator is proposed for automatically selecting the number of mixture components. In the present paper, a collection of univariate densities whose logarithm is locally {\beta}-H\"older with moment and tail conditions are considered. We show that this penalized estimator is minimax adaptive to the {\beta} regularity of such densities in the Hellinger sense.