On nonparametric estimation of density level sets

• 成果类型: Article
• 署名作者: Tsybakov, AB
• 署名单位: Sorbonne Universite; Russian Academy of Sciences
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1069362732
• 发表日期: 1997
• 页码: 948-969
• 关键词: boundaries; contour

摘要

Let X-1,...,X-n be independent identically distributed observations from an unknown probability density f((.)). Consider the problem of estimating the level set G = G(f)(lambda)= (x epsilon R-2: f(x) greater than or equal to lambda) from the sample X-1,...,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.