On nonparametric tests of positivity/monotonicity/convexity

• 成果类型: Article
• 署名作者: Juditsky, A; Nemirovski, A
• 署名单位: Technion Israel Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1021379863
• 发表日期: 2002
• 页码: 498-527
• 关键词: squared density derivatives; integral functionals; efficient estimation; CONVERGENCE; regression; hypothesis; Wavelets; signal

摘要

We consider the problem of estimating the distance from an unknown signal, observed in a white-noise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Holder class, the risk of estimating the L-r-distance, 1 less than or equal to r < infinity, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity, monotonicity and convexity of the unknown signal. We also provide an estimate for the distance to the cone of positive functions for which risk is, by a logarithmic factor, smaller than that of the plug-in estimate.