On Nonparametric Estimation of Density Level Sets

Let X1, ..., Xnbe independent identically distributed observations from an unknown probability density f(·). Consider the problem of estimating the level set G = Gf(λ) = {x ∈ R2: f(x) ≥ λ} from the sample X1, ..., Xn, under the assumption that the boundary of G has a certain smoothness. We propose p...

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Bibliographic Details
Published in:The Annals of statistics 1997-06, Vol.25 (3), p.948-969
Main Author: Tsybakov, A. B.
Format: Article
Language:English
Subjects:
62G05
62G20
Average linear density
Consistent estimators
Density
Density estimation
Density level set
Estimators
excess mass
Integers
Lebesgue measures
Mathematical functions
Nonparametric Function Estimation
optimal rate of convergence
piecewise-polynomial estimator
shape function
Shape functions
Spherical coordinates
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Summary:Let X1, ..., Xnbe independent identically distributed observations from an unknown probability density f(·). Consider the problem of estimating the level set G = Gf(λ) = {x ∈ R2: f(x) ≥ λ} from the sample X1, ..., Xn, 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.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1069362732