Strong identifiability and optimal minimax rates for finite mixture estimation

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 [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.