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：随着技术的飞速发展，多元函数型数据（MFD）在生物学、气候学、金融学等多个学科领域日益普及。尽管MFD广泛应用于各领域，但其均值函数假设检验方法，尤其是广义线性假设检验（GLHT）问题的研究仍十分有限。本研究针对MFD的GLHT问题提出并研究了一种新的全局检验方法，该方法将单因素函数型多变量方差分析（FMANOVA）、事后检验和对比分析作为特例。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。该卡方型混合分布可通过三累积量匹配的卡方近似方法进行良好逼近，其近似参数由数据估计得到。通过引入调整系数，所提检验方法无论函数型数据的相关结构如何，即使样本量相对较小，都能有效运行。此外，该方法被证明具有根号n一致性，即对局部备择假设具有非平凡检验功效。模拟研究和实际数据分析验证了所提方法在有限样本下的性能及其广泛适用性。