  In this paper, we study distance covariance, Hilbert-Schmidt covariance (aka
Hilbert-Schmidt independence criterion [Gretton et al. (2008)]) and related
independence tests under the high dimensional scenario. We show that the sample
distance/Hilbert-Schmidt covariance between two random vectors can be
approximated by the sum of squared componentwise sample cross-covariances up to
an asymptotically constant factor, which indicates that the
distance/Hilbert-Schmidt covariance based test can only capture linear
dependence in high dimension. As a consequence, the distance correlation based
t-test developed by Szekely and Rizzo (2013) for independence is shown to have
trivial limiting power when the two random vectors are nonlinearly dependent
but component-wisely uncorrelated. This new and surprising phenomenon, which
seems to be discovered for the first time, is further confirmed in our
simulation study. As a remedy, we propose tests based on an aggregation of
marginal sample distance/Hilbert-Schmidt covariances and show their superior
power behavior against their joint counterparts in simulations. We further
extend the distance correlation based t-test to those based on Hilbert-Schmidt
covariance and marginal distance/Hilbert-Schmidt covariance. A novel unified
approach is developed to analyze the studentized sample
distance/Hilbert-Schmidt covariance as well as the studentized sample marginal
distance covariance under both null and alternative hypothesis. Our theoretical
and simulation results shed light on the limitation of distance/Hilbert-Schmidt
covariance when used jointly in the high dimensional setting and suggest the
aggregation of marginal distance/Hilbert-Schmidt covariance as a useful
alternative.
