A global test for heteroscedastic one-way FMANOVA with applications

Multivariate functional data are prevalent in various fields such as biology, climatology, and finance. Motivated by the World Health Data applications, in this study, we propose and examine a global test for assessing the equality of multiple mean functions in multivariate functional data. This test addresses the one-way Functional Multivariate Analysis of Variance (FMANOVA) problem, which is a fundamental issue in the analysis of multivariate functional data. While numerous analysis of variance tests have been proposed and studied for univariate functional data, only a limited number of methods have been developed for the one-way FMANOVA problem. Furthermore, our global test has the ability to handle heteroscedasticity in the unknown covariance function matrices that underlie the multivariate functional data, which is not possible with existing methods. We establish the asymptotic null distribution of the test statistic as a chi-squared-type mixture, which depends on the eigenvalues of the covariance function matrices. To approximate the null distribution, we introduce a Welch–Satterthwaite type chi-squared-approximation with consistent parameter estimation. The proposed test exhibits root-$n$ consistency, meaning it possesses nontrivial power against a local alternative. Additionally, it offers superior computational efficiency compared to several permutation-based tests. Through simulation studies and applications to the World Health Data, we highlight the advantages of our global test.