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Serial and nonserial sign-and-rank statistics. Asymptotic representation and asymptotic normality

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  • Hallin, M.
  • Vermandele, C.
  • Werker, B.J.M.

    (Tilburg University, School of Economics and Management)

Abstract

The classical theory of rank-based inference is entirely based either on ordinary ranks, which do not allow for considering location nor intercept parameters, or on signed ranks, which require an assumption of symmetry.If the median, in the absence of a symmetry assumption, is considered as a location parameter, the maximal invariance property of ordinary ranks is lost to the ranks and the signs.As shown in Hallin and Werker (2003), conditioning on a maximal invariant in such situations is essential if semiparametric efficiency is to be reached.This new maximal invariant thus suggests a new class of statistics, based on ordinary ranks and signs.An asymptotic representation theory a la H ajek is developed here for such statistics, both in the nonserial and in the serial case.The corresponding asymptotic normality results clearly show how the signs are adding a separate contribution to the asymptotic variance, hence, potentially, to asymptotic e ciency.Applications to semiparametric inference in regression and time series models with median restrictions are treated in detail in a companion paper (Hallin, Werker, and Vermandele 2003).
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Suggested Citation

  • Hallin, M. & Vermandele, C. & Werker, B.J.M., 2006. "Serial and nonserial sign-and-rank statistics. Asymptotic representation and asymptotic normality," Other publications TiSEM 343e49a2-4527-4c03-b247-9, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:343e49a2-4527-4c03-b247-958d4adda80f
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    References listed on IDEAS

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    1. Marc Hallin & Madan Lal Puri, 1994. "Aligned rank tests for linear models with autocorrelated errors," ULB Institutional Repository 2013/2045, ULB -- Universite Libre de Bruxelles.
    2. Hallin, Marc & Puri, Madan L., 1991. "Time series analysis via rank order theory: Signed-rank tests for ARMA models," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 1-29, October.
    3. Horowitz J.L. & Spokoiny V.G., 2002. "An Adaptive, Rate-Optimal Test of Linearity for Median Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 822-835, September.
    4. Jean‐Marie Dufour, 1981. "Rank Tests For Serial Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 2(3), pages 117-128, May.
    5. Marc Hallin & Madan Lal Puri, 1988. "Optimal rank-based procedures for time series analysis: testing an ARMA model against other ARMA models," ULB Institutional Repository 2013/2013, ULB -- Universite Libre de Bruxelles.
    6. Marc Hallin & Madan Lal Puri, 1992. "Rank tests for time-series analysis: a survey," ULB Institutional Repository 2013/2229, ULB -- Universite Libre de Bruxelles.
    7. Marc Hallin & Guy Melard, 1988. "Rank-based tests for randomness against first-order serial dependence," ULB Institutional Repository 2013/2015, ULB -- Universite Libre de Bruxelles.
    8. Marc Hallin & Bas Werker, 1998. "Optimal testing for semiparametric autoregressive models: from Gaussian Lagrange multipliers to regression rank scores and adaptive tests," ULB Institutional Repository 2013/2219, ULB -- Universite Libre de Bruxelles.
    9. Hallin, M. & Puri, M. L., 1994. "Aligned Rank Tests for Linear Models with Autocorrelated Error Terms," Journal of Multivariate Analysis, Elsevier, vol. 50(2), pages 175-237, August.
    10. Marc Hallin & Catherine Vermandele & Bas J. M. Werker, 2008. "Semiparametrically efficient inference based on signs and ranks for median‐restricted models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(2), pages 389-412, April.
    11. Zhao, Quanshui, 2001. "Asymptotically Efficient Median Regression In The Presence Of Heteroskedasticity Of Unknown Form," Econometric Theory, Cambridge University Press, vol. 17(4), pages 765-784, August.
    12. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
    13. 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.
    14. Dufour, J.M., 1979. "Rank Tests for Serial Dependence," Cahiers de recherche 7815, Universite de Montreal, Departement de sciences economiques.
    15. Koenker, Roger, 2000. "Galton, Edgeworth, Frisch, and prospects for quantile regression in econometrics," Journal of Econometrics, Elsevier, vol. 95(2), pages 347-374, April.
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    Cited by:

    1. Andreou, Elena & Werker, Bas J.M., 2015. "Residual-based rank specification tests for AR–GARCH type models," Journal of Econometrics, Elsevier, vol. 185(2), pages 305-331.
    2. Marc Hallin & Catherine Vermandele & Bas J. M. Werker, 2008. "Semiparametrically efficient inference based on signs and ranks for median‐restricted models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(2), pages 389-412, April.
    3. Elena Andreou & Bas J.M. Werker, 2014. "Residual-based Rank Specification Tests for AR-GARCH type models," University of Cyprus Working Papers in Economics 02-2014, University of Cyprus Department of Economics.
    4. Werker, Bas J M & Andreou, Elena, 2013. "Residual-based Rank Specification Tests for AR-GARCH type models," CEPR Discussion Papers 9583, C.E.P.R. Discussion Papers.

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