Asymptotically most powerful rank tests for multivariate randomness against serial dependence

A class of linear serial multirank statistics is introduced for the problem of testing the null hypothesis that a multivariate series of observations is white noise (with unspecified density function) against alternatives of ARMA dependence. The asymptotic distributional properties of these statistics are investigated, both under the null as well as local alternative hypotheses. These statistics are shown to provide permutationally distribution-free tests that are asymptotically most powerful against specified local alternatives of ARMA dependence. In particular, a test of the van der Waerden type is shown to be asymptotically as powerful as the corresponding normal theory parametric test, based on classical sample autocovariances.