  This paper adresses the problem of testing for the equality of $k$
probability distributions on Hilbert spaces, with $k\geqslant 2$. We introduce
a generalization of the maximum variance discrepancy called multiple maximum
variance discrepancy (MMVD). Then, a consistent estimator of this measure is
proposed as test statistic, and its asymptotic distribution under the null
hypothesis is derived. A simulation study comparing the proposed test with
existing ones is provided
