The Annals of Statistics

On nonparametric estimation of density level sets

A. B. Tsybakov

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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

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

First available in Project Euclid: 20 November 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

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


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

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