  The kernel two-sample test based on the maximum mean discrepancy (MMD) is one
of the most popular methods for detecting differences between two distributions
over general metric spaces. In this paper we propose a method to boost the
power of the kernel test by combining MMD estimates over multiple kernels using
their Mahalanobis distance. We derive the asymptotic null distribution of the
proposed test statistic and use a multiplier bootstrap approach to efficiently
compute the rejection region. The resulting test is universally consistent and,
since it is obtained by aggregating over a collection of kernels/bandwidths, is
more powerful in detecting a wide range of alternatives in finite samples. We
also derive the distribution of the test statistic for both fixed and local
contiguous alternatives. The latter, in particular, implies that the proposed
test is statistically efficient, that is, it has non-trivial asymptotic
(Pitman) efficiency. Extensive numerical experiments are performed on both
synthetic and real-world datasets to illustrate the efficacy of the proposed
method over single kernel tests. Our asymptotic results rely on deriving the
joint distribution of MMD estimates using the framework of multiple stochastic
integrals, which is more broadly useful, specifically, in understanding the
efficiency properties of recently proposed adaptive MMD tests based on kernel
aggregation.
