Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence

Abstract

We propose a multivariate generalization of signed-rank tests for testing elliptically symmetric white noise against ARMA serial dependence. These tests are based on Randles's concept of interdirections and the ranks of pseudo-Mahalanobis distances. They are affine-invariant and asymptotically equivalent to strictly distribution-free statistics. Depending on the score function considered (van der Waerden, Laplace. $\ldots$), they allow for locally asymptotically maximin tests at selected densities (multivariate normal, multivariate double exponential, $\ldots$). Local powers and asymptotic relative efficiencies with respect to the Gaussian procedure are derived. We extend to the multivariate serial context the Chernoff--Savage result, showing that classical correlogram-based procedures are uniformly dominated by the van der Waerden version of our tests, so that correlogram methods are not admissible in the Pitman sense. We also prove an extension of the celebrated Hodges--Lehmann `.864 result', providing, for any fixed space dimension, the lower bound for the asymptotic relative efficiency of the proposed multivariate Spearman type tests with respect to the Gaussian tests. These asymptotic results are confirmed by a Monte Carlo investigation.