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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 m₀ components, the optimal local minimax rate of estimation of a mixing distribution with m components is n^-1/ (4 (m-m₀) +2). This corrects a previous paper by Chen Ann. Statist. 23 (1995) 221–233. By contrast, it turns out that there are estimators with a (nonuniform) pointwise rate of estimation of n^-1/2 for all mixing distributions with a finite number of components.
Heinrich et al. (Fri,) studied this question.