1. Strong identifiability and optimal minimax rates for finite mixture estimation
- Author
-
Jonas Kahn, Philippe Heinrich, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), and Laboratoire Paul Painlevé - UMR 8524 (LPP)
- Subjects
Statistics and Probability ,Local asymptotic normality ,Wasserstein metric ,strong identifiability ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,superefficiency ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,0202 electrical engineering, electronic engineering, information engineering ,mixing distribution ,Applied mathematics ,62G05 ,convergence of experiments ,0101 mathematics ,Mixing (physics) ,62G20 ,Mathematics ,Pointwise ,mixture model ,Estimator ,020206 networking & telecommunications ,pointwise rate ,Minimax ,maximum likelihood estimate ,Distribution (mathematics) ,Rate of convergence ,Identifiability ,Statistics, Probability and Uncertainty ,rate of convergence - 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 [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.
- Published
- 2018