Nonparametric estimation of the mixing density using polynomials

TL;DR

This paper introduces an orthogonal series estimator using Legendre polynomials for nonparametric mixing density estimation from i.i.d. data, providing minimax bounds and adaptivity results across various mixture models.

Contribution

It develops a novel Legendre polynomial-based orthogonal series estimator for mixing densities, with theoretical minimax bounds and adaptivity in exponential, Gamma, and Beta mixture models.

Findings

Estimator achieves minimax rate with m ~ A log(n) in exponential mixtures.

Provides adaptive estimation over smoothness classes.

Offers a consistent estimator for the support of the mixing density.

Abstract

We consider the problem of estimating the mixing density $f$ from $n$ i.i.d. observations distributed according to a mixture density with unknown mixing distribution. In contrast with finite mixtures models, here the distribution of the hidden variable is not bounded to a finite set but is spread out over a given interval. We propose an approach to construct an orthogonal series estimator of the mixing density $f$ involving Legendre polynomials. The construction of the orthonormal sequence varies from one mixture model to another. Minimax upper and lower bounds of the mean integrated squared error are provided which apply in various contexts. In the specific case of exponential mixtures, it is shown that the estimator is adaptive over a collection of specific smoothness classes, more precisely, there exists a constant $A\textgreater{}0$ such that, when the order $m$ of the projection estimator verifies $m\sim A \log(n)$, the estimator achieves the minimax rate over this collection. Other cases are investigated such as Gamma shape mixtures and scale mixtures of compactly supported densities including Beta mixtures. Finally, a consistent estimator of the support of the mixing density $f$ is provided.