Optimal rates for finite mixture estimation

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.

48 pages, 1 figure, submitted to The Annals of Statistics, as a main article (30 pages) and the appendices (19 pages) as supplemental material. Part of the material appears in an earlier version appears as arXiv:1504.03506, but without any result on pointwise rates, any figure, much less bibliography and explanations, and overall different presentation