OPTIMAL RATES FOR FINITE MIXTURE ESTIMATION

Philippe Heinrich (1) , Jonas Kahn (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

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 \citet{Chen} 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.

Mots clés: strong identifiability.; rate of convergence; mixture model; mixing distribution; Wasserstein metric; maximum likelihood estimate; convergence of experiments; Local asymptotic normality; pointwise rate; Superefficient estimator

Domaines: Statistiques [math.ST]

Philippe Heinrich, Jonas Kahn. OPTIMAL RATES FOR FINITE MIXTURE ESTIMATION. 2015. ⟨hal-01176844⟩