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 recall the notion of $\gamma$-exponent of a density at a certain level. This notion is similar to Tsybakov's margin assumption and allows us to prove fast rates of convergence for general plug-in methods, up to order $n^{-1}$ when the density is supposed to be smooth in a neighborhood of the level under consideration. Lower bounds proving optimality of the rates in a minimax sense are also provided. Finally, when the density jumps around the level under consideration, we show that exponential rates of convergence are attainable.

Mots clés: Density level set; plug-in estimators; fast rates of convergence; minimax lower bounds

Domaines: Statistiques [math.ST]

Philippe Rigollet, Régis Vert. Fast rates for plug-in estimators of density level sets. 2006. ⟨hal-00114180v1⟩