A powerful test based on tapering for use in functional data analysis

TL;DR

This paper introduces a tapering-based test for functional linear models that emphasizes smooth attributes by down-weighting higher Fourier frequencies, achieving adaptive optimality and better non-asymptotic power in practical applications.

Contribution

It proposes a new tapering-based test statistic with adaptive optimality for functional data analysis, improving power against relevant alternatives.

Findings

The test emphasizes smoothness by down-weighting high frequencies.

It achieves adaptive optimality among tapering-based tests.

The test shows better non-asymptotic power in practical scenarios.

Abstract

A test based on tapering is proposed for use in testing a global linear hypothesis under a functional linear model. The test statistic is constructed as a weighted sum of squared linear combinations of Fourier coefficients, a tapered quadratic form, in which higher Fourier frequencies are down-weighted so as to emphasize the smooth attributes of the model. A formula is $Q_n^{OPT}=n\sum_{j=1}^{p_n}j^{-1/2}\|\boldsymbol{Y}_{n,j}\|^2$. Down-weighting by $j^{-1/2}$ is selected to achieve adaptive optimality among tests based on tapering with respect to its ``rates of testing,'' an asymptotic framework for measuring a test's retention of power in high dimensions under smoothness constraints. Existing tests based on truncation or thresholding are known to have superior asymptotic power in comparison with any test based on tapering; however, it is shown here that high-order effects can be substantial, and that a test based on $Q_n^{OPT}$ exhibits better (non-asymptotic) power against the sort of alternatives that would typically be of concern in functional data analysis applications. The proposed test is developed for use in practice, and demonstrated in an example application.