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June 1997 On nonparametric estimation of density level sets
A. B. Tsybakov
Ann. Statist. 25(3): 948-969 (June 1997). DOI: 10.1214/aos/1069362732


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.


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A. B. Tsybakov. "On nonparametric estimation of density level sets." Ann. Statist. 25 (3) 948 - 969, June 1997.


Published: June 1997
First available in Project Euclid: 20 November 2003

zbMATH: 0881.62039
MathSciNet: MR1447735
Digital Object Identifier: 10.1214/aos/1069362732

Primary: 62G05 , 62G20

Keywords: Density level set , excess mass , Optimal rate of convergence , piecewise-polynomial estimator , shape function

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • June 1997
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