We consider an ANOVA type multi-sample comparison problem, where the observations are infinite dimensional in nature. In particular, the observations may be random functions. We propose a test based on spatial signs in such a setup. An asymptotic implementation and a bootstrap implementation of this test are developed, and the asymptotic consistency of both procedures is established. We compare the finite-sample and asymptotic performances of our test with that of several other tests of ANOVA for functional data in the literature. We found that our test not only outperforms those tests in several non-Gaussian models with heavy tails, but in some Gaussian models also, it exhibits better performance than them.
