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December 2004 Rank-based optimal tests of the adequacy of an elliptic VARMA model
Marc Hallin, Davy Paindaveine
Ann. Statist. 32(6): 2642-2678 (December 2004). DOI: 10.1214/009053604000000724


We are deriving optimal rank-based tests for the adequacy of a vector autoregressive-moving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudo-Mahalanobis distances and on normed residuals computed from Tyler’s [Ann. Statist. 15 (1987) 234–251] scatter matrix; they generalize the univariate signed rank procedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1–29]. Two types of optimality properties are considered, both in the local and asymptotic sense, a la Le Cam: (a) (fixed-score procedures) local asymptotic minimaxity 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 innovation densities, including those with infinite variance and heavy-tails. We show that the AREs of our fixed-score procedures, with respect to traditional (Gaussian) methods, are the same as for the tests of randomness proposed in Hallin and Paindaveine [Bernoulli 8 (2002b) 787–815]. The multivariate serial extensions of the classical Chernoff–Savage and Hodges–Lehmann results obtained there thus also hold here; in particular, the van der Waerden versions of our tests are uniformly more powerful than those based on cross-covariances. As for our estimated-score procedures, they are fully adaptive, hence, uniformly optimal over the class of innovation densities satisfying the required technical assumptions.


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Marc Hallin. Davy Paindaveine. "Rank-based optimal tests of the adequacy of an elliptic VARMA model." Ann. Statist. 32 (6) 2642 - 2678, December 2004.


Published: December 2004
First available in Project Euclid: 7 February 2005

zbMATH: 1076.62044
MathSciNet: MR2153998
Digital Object Identifier: 10.1214/009053604000000724

Primary: 62G10 , 62M10

Keywords: Elliptical symmetry , local asymptotic normality , multivariate ranks , VARMA models

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 6 • December 2004
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