## The Annals of Statistics

### Minimax adaptive tests for the functional linear model

#### Abstract

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.

#### Article information

Source
Ann. Statist. Volume 41, Number 2 (2013), 838-869.

Dates
First available in Project Euclid: 29 May 2013

https://projecteuclid.org/euclid.aos/1369836962

Digital Object Identifier
doi:10.1214/13-AOS1093

Mathematical Reviews number (MathSciNet)
MR3099123

Zentralblatt MATH identifier
1267.62059

Subjects
Primary: 62J05: Linear regression
Secondary: 62G10: Hypothesis testing

#### Citation

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.

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