Optimal rates for finite mixture estimation

TL;DR

This paper establishes the optimal local minimax rates for estimating finite mixture distributions, correcting previous results, and highlights the existence of estimators achieving parametric rates under certain conditions.

Contribution

It provides the first precise characterization of the optimal rates for finite mixture estimation under regularity and identifiability assumptions.

Findings

Optimal local minimax rate is $n^{-1/(4(m-m_0)+2)}$ for estimating mixing distributions.

Existence of estimators with $n^{-1/2}$ pointwise rate for finite mixture models.

Correction of previous results by Chen (1995) regarding estimation rates.

Abstract

We study the rates of estimation of finite mixing distributions, that is, the parameters of the mixture. We prove that under some regularity and strong identifiability conditions, around a given mixing distribution with $m_0$ components, the optimal local minimax rate of estimation of a mixing distribution with $m$ components is $n^{-1/(4(m-m_0) + 2)}$. This corrects a previous paper by Chen (1995) in The Annals of Statistics. By contrast, it turns out that there are estimators with a (non-uniform) pointwise rate of estimation of $n^{-1/2}$ for all mixing distributions with a finite number of components.