The Annals of Statistics
- Ann. Statist.
- Volume 41, Number 2 (2013), 838-869.
Minimax adaptive tests for the functional linear model
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.
Ann. Statist. Volume 41, Number 2 (2013), 838-869.
First available in Project Euclid: 29 May 2013
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Hilgert, Nadine; Mas, André; Verzelen, Nicolas. Minimax adaptive tests for the functional linear model. Ann. Statist. 41 (2013), no. 2, 838--869. doi:10.1214/13-AOS1093. https://projecteuclid.org/euclid.aos/1369836962.
- Supplementary material: Technical Appendix. We provide additional control of the power and we describe the remaining proofs.