A median test for functional data

The median test has been proven to be more powerful than the Student t-test and the Wilcoxon–Mann–Whitney test in heavy-tailed cases for univariate data. The multivariate extension of the median test, for multidimensional data, was demonstrated to be more efficient than the Hotelling T2 and the Wilcoxon–Mann–Whitney tests for high dimensions and in very heavy-tailed cases. On the basis of these postulates, in this paper, we construct a median-type test based on spatial ranks for functional data, i.e. in infinite-dimensional space, and we obtain asymptotic results. Then, we compare the proposed functional median test with numerous competing tests using simulated and real functional data: as in the univariate and multivariate cases, the proposed test is more adapted to heavy-tailed distributions.