GEOMETRIZING RATES OF CONVERGENCE .2.

• 成果类型: Article
• 署名作者: DONOHO, DL; LIU, RC
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176348114
• 发表日期: 1991
• 页码: 633-667
• 关键词: DENSITY-ESTIMATION; minimax tests; kernel; Capacities; models; point; noise

摘要

Consider estimating a functional T(F) of an unknown distribution F member-of F from data X1,...,X(n) i.i.d. F. Let omega(epsilon) denote the modulus of continuity of the functional T over F, computed with respect to Hellinger distance. For well-behaved loss functions l(t), we show that inf(T)n sup(F) E(F)l(T(n) - T(F)) is equivalent to l(omega(n-1/2)) to within constants, whenever T is linear and F is convex. The same conclusion holds in three nonlinear cases: estimating the rate of decay of a density, estimating the mode and robust nonparametric regression. We study the difficulty of testing between the composite, infinite dimensional hypotheses H0: T(F) less-than-or-equal-to t and H1: T(F) greater-than-or-equal-to t + DELTA. Our results hold, in the cases studied, because the difficulty of the full infinite-dimensional composite testing problem is comparable to the difficulty of the hardest simple two-point testing subproblem.