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We are deriving optimal rank-based tests for the adequacy of a vectorautoregressive-moving average (VARMA) model with elliptically contouredinnovation density. These tests are based on the ranks of pseudo-Mahalanobisdistances and on normed residuals computed from Tyler's [Ann. Statist. 15(1987) 234-251] scatter matrix; they generalize the univariate signed rankprocedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1-29].Two types of optimality properties are considered, both in the local andasymptotic sense, a la Le Cam: (a) (fixed-score procedures) local asymptoticminimaxity at selected radial densities, and (b) (estimated-score procedures)local asymptotic minimaxity uniform over a class F of radial densities.Contrary to their classical counterparts, based on cross-covariance matrices,these tests remain valid under arbitrary elliptically symmetric innovationdensities, including those with infinite variance and heavy-tails. We show thatthe AREs of our fixed-score procedures, with respect to traditional (Gaussian)methods, are the same as for the tests of randomness proposed in Hallin andPaindaveine [Bernoulli 8 (2002b) 787-815]. The multivariate serial extensionsof the classical Chernoff-Savage and Hodges-Lehmann results obtained there thusalso hold here; in particular, the van der Waerden versions of our tests areuniformly more powerful than those based on cross-covariances. As for ourestimated-score procedures, they are fully adaptive, hence, uniformly optimalover the class of innovation densities satisfying the required technicalassumptions.

Rank-based optimal tests of the adequacy of an elliptic VARMA model