As technology continues to advance at a rapid pace, the prevalence of multivariate functional data (MFD) has expanded across diverse disciplines, spanning biology, climatology, finance, and numerous other fields of study. Although MFD are encountered in various fields, the development of methods for hypotheses on mean functions, especially the general linear hypothesis testing (GLHT) problem for such data has been limited. In this study, we propose and study a new global test for the GLHT problem for MFD, which includes the one-way FMANOVA, post hoc, and contrast analysis as special cases. The asymptotic null distribution of the test statistic is shown to be a chi-squared-type mixture dependent of eigenvalues of the heteroscedastic covariance functions. The distribution of the chi-squared-type mixture can be well approximated by a three-cumulant matched chi-squared-approximation with its approximation parameters estimated from the data. By incorporating an adjustment coefficient, the proposed test performs effectively irrespective of the correlation structure in the functional data, even when dealing with a relatively small sample size. Additionally, the proposed test is shown to be root-n consistent, that is, it has a nontrivial power against a local alternative. Simulation studies and a real data example demonstrate finite-sample performance and broad applicability of the proposed test.

翻译：随着技术的快速发展，多元函数型数据在生物学、气候学、金融学等多个学科领域的应用日益广泛。尽管这些数据在多个领域均有涉及，但关于均值函数的假设检验方法，尤其是针对函数型数据的一般线性假设检验问题的研究仍十分有限。本研究提出并探讨了一种针对多元函数型数据一般线性假设检验问题的全局检验方法，该方法将单因素函数型多元方差分析、事后检验和对照分析作为特例。检验统计量的渐近零分布被证明是依赖于异方差协方差函数特征值的卡方型混合分布。该混合分布可通过从数据中估计的三累积量匹配卡方近似进行有效逼近。通过引入调整系数，所提出的检验方法无论函数型数据中是否存在相关结构，即使样本量相对较小也能有效运行。此外，该方法被证明是根号n一致的，即具有对局部备择假设的非平凡检验势。模拟研究和实际数据示例验证了该方法的有限样本性能及广泛适用性。