MINIMAX ADAPTIVE TESTS FOR THE FUNCTIONAL LINEAR MODEL

• 成果类型: Article
• 署名作者: Hilgert, Nadine; Mas, Andre; Verzelen, Nicolas
• 署名单位: INRAE; Institut Agro; Montpellier SupAgro; Universite de Montpellier
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1093
• 发表日期: 2013
• 页码: 838-869
• 关键词: principal-components-analysis; regression models; estimators; hypotheses; rates

摘要

We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional principal component analysis. Interestingly, the procedures are completely data-driven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local alternatives are assessed in a nonasymptotic setting. This allows us to prove that these procedures are minimax adaptive (up to an unavoidable log log n multiplicative term) to the unknown regularity of the slope. As a side result, the minimax separation distances of the slope are derived for a large range of regularity classes. A numerical study illustrates these theoretical results.