Covariance Test for Discretely Observed Functional Data: When and How it Works?

• 成果类型: Article; Early Access
• 署名作者: Zhou, Yang; Yang, Jin; Yao, Fang
• 署名单位: Beijing Normal University; Hong Kong Polytechnic University; Peking University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.2026.2641240
• 发表日期: 2026-05-27
• 关键词: Diverging truncation; Functional principal components; perturbation bounds; Phase Transition; Principal Component Analysis; linear-regression; convergence-rates; Operators; EQUALITY; prediction; SPARSE

摘要

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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.