### On nonparametric estimation of density level sets

A. B. Tsybakov
Source: Ann. Statist. Volume 25, Number 3 (1997), 948-969.

#### Abstract

Let $X_1, \dots, X_n$ be independent identically distributed observations from an unknown probability density $f(\cdot)$. Consider the problem of estimating the level set $G = G_f(\lambda) = {x \epsilon\mathbb{R}^2: f(x) \geq \lambda}$ from the sample $X_1, \dots, X_n$, under the assumption that the boundary of G has a certain smoothness. We propose piecewise-polynomial estimators of G based on the maximization of local empirical excess masses. We show that the estimators have optimal rates of convergence in the asymptotically minimax sense within the studied classes of densities. We find also the optimal convergence rates for estimation of convex level sets. A generalization to the N-dimensional case, where $N > 2$, is given.

First Page:
Primary Subjects: 62G05, 62G20
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aos/1069362732
Mathematical Reviews number (MathSciNet): MR1447735
Digital Object Identifier: doi:10.1214/aos/1069362732
Zentralblatt MATH identifier: 0881.62039

2013 © Institute of Mathematical Statistics

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