Strong identifiability and optimal minimax rates for finite mixture estimation (Q1991679)

Excerpt from abstract and paper: ``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 \textit{J. Chen} [Ann. Stat. 23, No. 1, 221--233 (1995; Zbl 0821.62023)]. [\dots] The rate gets worse with more components, which is consistent with the behaviour when there are infinitely many components, such as deconvolution. [...] In addition, the optimal local minimax rate and the optimal pointwise rate of estimation everywhere are not the same. This discrepancy is unusual in statistics, and probably the reason why the $n^{-1/4}$ rate went unchallenged for twenty years. Specifically, if instead of comparing all pairs of mixtures in a ball, we allow only one mixture in it, we get (21) which corrects Lemma 2 of Chen [loc. cit.]. As a consequence, Theorem 2 of Chen is valid by dropping uniformity: for any fixed mixing distribution say $G$, the estimator considered there will converge at rate $n^{-1/4}$, but with a multiplicative constant that depends on $G$. It then becomes a statement on the optimal pointwise rate of estimation everywhere, and can even be strengthened to $n^{-1/2}$.''