A Modified Test for Linear Hypotheses in Multivariate Functional Data Under Heteroscedasticity and Small Samples

As big data continues to grow, statistical inference for multivariate functional data (MFD) has become increasingly important. Although recent advancements have been made in testing the equality of mean functions, research on testing linear hypotheses for mean functions remains limited. Current methods primarily consist of resampling-based tests or asymptotic tests. However, resampling-based tests are known to be time-consuming, while asymptotic tests typically require larger sample sizes to maintain accurate Type I error control. This paper introduces a finite-sample test that modifies the traditional Wilks’ lambda test from MANOVA to address general linear hypothesis testing for MFD. The test statistic is based on two symmetric, nonnegative-definite matrices, which are approximated by Wishart distributions, with degrees of freedom estimated via a U-statistics-based approach. The proposed test is affine-invariant, robust to heteroscedasticity, computationally more efficient than resampling-based tests, and better at controlling significance levels in small samples compared with asymptotic tests. A real-data example illustrates the practical utility of the method.