The Annals of Statistics

On nonparametric estimation of density level sets

A. B. Tsybakov

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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.

Article information

Source
Ann. Statist., Volume 25, Number 3 (1997), 948-969.

Dates
First available in Project Euclid: 20 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1069362732

Digital Object Identifier
doi:10.1214/aos/1069362732

Mathematical Reviews number (MathSciNet)
MR1447735

Zentralblatt MATH identifier
0881.62039

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties

Keywords
Density level set excess mass shape function optimal rate of convergence piecewise-polynomial estimator

Citation

Tsybakov, A. B. On nonparametric estimation of density level sets. Ann. Statist. 25 (1997), no. 3, 948--969. doi:10.1214/aos/1069362732. https://projecteuclid.org/euclid.aos/1069362732


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