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

By Marc Hallin, Davy Paindaveine. 2004

61 citations recorded by the source corpus.

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 narmed 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 assomptions.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/00ec3550ce53aa46a8f40fbf68fb610bd280a147

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Citations in the corpus

Monge-Kantorovich Depth, Quantiles, Ranks and Signs (2014, background): “There, elliptical contours with P -probability contents τ provide a natural definition of τ -quantile contours, while the ranks and unit vectors associated with sphericized observations have proven to be adequate concepts of multivariate ranks and signs, as shown in [19], [20], [21] and [22]: call t”
Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors (2005, background): “The nonparametric (signed rank) J-score versions of the serial and nonserial statistics (7) and ( 8 ) are, in the serial case,”
ECA: High-Dimensional Elliptical Component Analysis in Non-Gaussian Distributions (2013, background): “Theoretical analysis in low dimensions is provided by Hallin and Paindaveine (2002a, 2002b, 2004, 2005, 2006), Hallin (Oja); Hallin, Paindaveine, and Verdebout (2010, 2014), and some new extensions to high-dimensional settings are provided in Croux, Filzmoser, and Fritz (2013) and Feng and Liu…”
SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE II. OPTIMAL R-ESTIMATION OF SHAPE (2006, unknown): “In such a context, robust inference methods, resisting arbitrarily heavy radial tails, are highly desirable and distribution-free rank-based methods naturally come into the picture (see [9, 10, 11, 12] for closely related results).”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, background): “…Bickel 1965; Puri and Sen 1966; spatial ranks, e.g., Chaudhuri 1996; Marden 1999; depth-based ranks, e.g., Liu and Singh 1993; Zuo and Serfling 2000; Mahalanobis ranks, e.g., Hallin and Paindaveine 2004, 2006) and the corresponding rank-based tests no longer possess exact distribution-freeness.”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, methodology): “…dependence (Hallin and Paindaveine 2002b), linear models with VARMA errors (Hallin and Paindaveine 2004a, 2005a, 2006a), VAR order identification (Hallin and Paindaveine 2004b), shape (Hallin and Paindaveine 2006b; Hallin, Oja, and Paindaveine 2006), homogeneity of scatter (Hallin and…”
OPTIMAL RANK-BASED TESTING FOR PRINCIPAL COMPONENTS (2010, methodology): “Although this powerful method has been used quite successfully in inference problems for elliptical families [Hallin and Paindaveine (2002, 2004, 2005, 2006a), Hallin, Oja and Paindaveine (2006) and Hallin and Paindaveine (2008a) for location, VARMA dependence, linear models, shape and scatter,…”