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On multivariate runs tests for randomness

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  • Davy Paindaveine

Abstract

This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999), and (ii) an original concept of matrix-valued runs. The resulting runs tests themselves exist in various versions, one of which is a function of the number of data-based hyperplanes separating pairs of observations only. All proposed multivariate runs tests are affine-invariant and highly robust: in particular, they allow for heteroskedasticity and do not require any moment assumption. Their limiting distributions are derived under the null hypothesis and under sequences of local vector ARMA alternatives. Asymptotic relative efficiencies with respect to Gaussian Portmanteau tests are computed, and show that, while Mardentype runs tests suffer severe consistency problems, tests based on matrix-valued runs perform uniformly well for moderate-to-large dimensions. A Monte-Carlo study confirms the theoretical results and investigates the robustness properties of the proposed procedures. A real data example is also treated, and shows that combining both types of runs tests may provide some insight on the reason why rejection occurs, hence that Marden-type runs tests, despite their lack of consistency, also are of interest for practical purposes.

Suggested Citation

  • Davy Paindaveine, 2009. "On multivariate runs tests for randomness," Working Papers ECARES 2009_002, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2009_002
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    References listed on IDEAS

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    1. Marc Hallin & Jean-François Ingenbleek & Madan Lal Puri, 1984. "Linear serial rank tests for randomness against ARMA alternatives," ULB Institutional Repository 2013/2167, ULB -- Universite Libre de Bruxelles.
    2. Peter Hall & J. S. Marron & Amnon Neeman, 2005. "Geometric representation of high dimension, low sample size data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 427-444.
    3. Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
    4. Taskinen, Sara & Kankainen, Annaliisa & Oja, Hannu, 2003. "Sign test of independence between two random vectors," Statistics & Probability Letters, Elsevier, vol. 62(1), pages 9-21, March.
    5. Denis Larocque & Jaakko Nevalainen & Hannu Oja, 2007. "A weighted multivariate sign test for cluster-correlated data," Biometrika, Biometrika Trust, vol. 94(2), pages 267-283.
    6. Lutz Dümbgen & David E. Tyler, 2005. "On the Breakdown Properties of Some Multivariate M-Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 247-264.
    7. Hallin, Marc & Paindaveine, Davy, 2005. "Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 122-163, March.
    8. Marc Hallin & Jean-Marie Dufour & Ivan Mizera, 1998. "Generalized run tests for heteroscedastic time series," ULB Institutional Repository 2013/2077, ULB -- Universite Libre de Bruxelles.
    9. Haataja, Riina & Larocque, Denis & Nevalainen, Jaakko & Oja, Hannu, 2009. "A weighted multivariate signed-rank test for cluster-correlated data," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1107-1119, July.
    10. Möttönen, J. & Hüsler, J. & Oja, H., 2003. "Multivariate nonparametric tests in a randomized complete block design," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 106-129, April.
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    Cited by:

    1. Rainer Dyckerhoff & Christophe Ley & Davy Paindaveine, 2014. "Depth-Based Runs Tests for bivariate Central Symmetry," Working Papers ECARES ECARES 2014-03, ULB -- Universite Libre de Bruxelles.
    2. Van Bever, Germain, 2016. "Simplicial bivariate tests for randomness," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 20-25.
    3. Ludwig Baringhaus & Norbert Henze, 2016. "Revisiting the two-sample runs test," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 432-448, September.
    4. Davy Paindaveine & Thomas Verdebout, 2013. "Universal Asymptotics for High-Dimensional Sign Tests," Working Papers ECARES ECARES 2013-40, ULB -- Universite Libre de Bruxelles.

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    Keywords

    elliptical distributions; interdirections;

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