Fast rates for plug-in estimators of density level sets

Philippe Rigollet (1) , Régis Vert (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 MASA - Masagroup

In the context of density level set estimation, we study the convergence of general plug-in methods under two main assumptions on the density for a given level $\lambda$. More precisely, it is assumed that the density (i) is smooth in a neighborhood of $\lambda$ and (ii) has $\gamma$-exponent at level $\lambda$. Condition (i) ensures that the density can be estimated at a standard nonparametric rate and condition (ii) is similar to Tsybakov's margin assumption which is stated for the classification framework. Under these assumptions, we derive optimal rates of convergence for plug-in estimators. Explicit convergence rates are given for plug-in estimators based on kernel density estimators when the underlying measure is the Lebesgue measure. Lower bounds proving optimality of the rates in a minimax sense when the density is Hölder smooth are also provided.

Mots clés: Density level sets; plug-in estimators; rates of convergence; kernel density estimators; minimax lower bounds.; minimax lower bounds

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Philippe Rigollet, Régis Vert. Fast rates for plug-in estimators of density level sets. 2008. ⟨hal-00114180v4⟩