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.

翻译：针对函数型数据分析中的协方差检验，现有方法仅适用于完全观测曲线，而实际中轨迹通常以离散形式观测并伴有噪声。为弥补这一差距，我们采用池化平滑策略构建基于FPC的检验统计量，允许估计特征函数的数量随样本量增长。这产生了一致性的非参数检验，而挑战源于发散截断与离散观测的并存。借助估计特征函数的先进扰动界，我们证明渐近零分布可在允许的截断水平下保持有效性。此外，当采样频率（即每个受试者的测量次数）达到样本量的特定量级时，检验表现如同函数被完全观测。这一相变现象不同于池化均值/协方差估计的已知结论，反映了协方差检验因特征分解而面临的更高难度。数值研究（包括模拟与真实数据示例）表明，相较于现有方法，本方法性能更优。