One-way MANOVA for functional data via Lawley–Hotelling trace test

Functional data arise from various fields of study and there have been numerous works on their analysis. However, most of existing methods consider the univariate case and methodology for multivariate functional data analysis is rather limited. In this article, we consider testing equality of vectors of mean functions for multivariate functional data, i.e., functional one-way multivariate analysis of variance (MANOVA). To this aim, we study asymptotic null distribution of the functional Lawley–Hotelling trace (FLH) test statistic and approximate it by a Welch–Satterthwaite type χ²-approximation. We describe two approaches to estimating the parameters in the χ²-approximation ratio-consistently. The resulting FLH test has the correct asymptotic level, is root-n consistent in detecting local alternatives, and is computationally efficient. The numerical performance is examined via some simulation studies and application to three real data examples. The proposed FLH test is comparable with four existing tests based on permutation in terms of size control and power. The major advantage is that it is much faster to compute.