Covariance test for discretely observed functional data: when and how it works?

TL;DR

This paper develops a nonparametric covariance test for discretely observed functional data, establishing its validity and phase transition behavior as sampling frequency increases, with demonstrated superior performance.

Contribution

It introduces a novel FPC-based test that remains valid under discretization and noise, advancing eigenfunction perturbation bounds and revealing a phase transition phenomenon.

Findings

Test maintains valid null distribution across truncation levels.

Test behaves as if functions are fully observed when sampling frequency is high.

Numerical studies show superior performance over existing methods.

Abstract

For covariance test in functional data analysis, existing methods are developed only for fully observed curves, whereas in practice, trajectories are typically observed discretely and with noise. To bridge this gap, we employ a pool-smoothing strategy to construct an FPC-based test statistic, allowing the number of estimated eigenfunctions to grow with the sample size. This yields a consistently nonparametric test, while the challenge arises from the concurrence of diverging truncation and discretized observations. Facilitated by advancing perturbation bounds of estimated eigenfunctions, we establish that the asymptotic null distribution remains valid across permissable truncation levels. Moreover, when the sampling frequency (i.e., the number of measurements per subject) reaches certain magnitude of sample size, the test behaves as if the functions were fully observed. This phase transition phenomenon differs from the well-known result of the pooling mean/covariance estimation, reflecting the elevated difficulty in covariance test due to eigen-decomposition. The numerical studies, including simulations and real data examples, yield favorable performance compared to existing methods.