Abstract:

Distance covariance and distance correlation are well known to completely characterize the independence between two random vectors of fixed dimensions. In this talk, we show that the sample distance covariance between two random vectors can be approximated by the sum of squared componentwise sample crosscovariances up to an asymptotically constant factor, which indicates that the distance 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 componentwisely uncorrelated. This new and surprising phenomenon, which seems to be discovered for the first time, is further confirmed in our simulation studies. The same phenomenon also holds for the HilbertSchmidt independence criterion in the high dimension low sample size setting. A remedy based on aggregation of marginal sample distance covariances is proposed and compared in the simulation studies.
