Minimax adaptive tests for the Functional Linear model

TL;DR

This paper introduces two data-driven, adaptive testing procedures for the functional linear model that do not require prior smoothness knowledge, achieving minimax optimality in detecting slope function nullity.

Contribution

The paper proposes novel, fully data-driven tests combining multiple testing and random projections, with proven minimax adaptivity to unknown regularity of the slope.

Findings

Tests are minimax adaptive up to a log-log factor.

Procedures perform well in nonasymptotic settings.

Numerical results confirm theoretical properties.

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.