Minimax adaptive tests for the Functional Linear model

Nadine Hilgert (1) , André Mas (2) , Nicolas Verzelen (1)

1 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier

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.

Mots clés: functional linear regression; principal component analysis; adaptive testing; minimax hypothesis testing; multiple testing; ellipsoid; goodness-of-fit

Mots clés: eigenfunction; minimax separation rate

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Nadine Hilgert, André Mas, Nicolas Verzelen. Minimax adaptive tests for the Functional Linear model. Annals of Statistics, 2013, 41 (2), pp.838-869. ⟨10.1214/13-AOS1093⟩. ⟨hal-00819528⟩