Optimal calibration for multiple testing against local inhomogeneity in higher dimension

• 成果类型: Article
• 署名作者: Rohde, Angelika
• 署名单位: University of Hamburg
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0263-1
• 发表日期: 2011
• 页码: 515-559
• 关键词: GOODNESS-OF-FIT; data-driven version; probability-inequalities; 2-sample problem; optimal recovery; random-variables; Rank test; sup-norm; replacement; regression

摘要

Based on two independent samples X (1), . . . , X (m) and X (m+1), . . . , X (n) drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearest-neighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The properly adjusted multiple testing procedure is shown to be sharp-optimal for typical arrangements of the observation values which appear with probability close to one. The proof relies on a new coupling Bernstein type exponential inequality, reflecting the non-subgaussian tail behavior of a combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis p = q to a simple one with the multivariate mixed density (m/n)p + (1 - m/n)q as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic Holder classes. The exact minimax risk asymptotics are obtained in terms of solutions of the optimal recovery.