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Linear And Quadratic Serial Rank Tests For Randomness Against Serial Dependence

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  • Marc. Hallin
  • Jean‐François Ingenbleek
  • Madan L. Puri

Abstract

. The problem of testing randomness against specified and unspecified contiguous ARMA alternatives is considered. In the case of specified alternative, the linear serial rank tests proposed by the authors (Hallin, Ingenbleek and Puri 1985) are shown to be asymptotically most powerful within the class of all possible tests (at the required level). In the case of unspecified alternative, however, any of the above optimal tests is unfortunately completely insensitive against a whole subclass of the alternative. Quadratic serial rank statistics, providing tests of the χ2‐type are therefore introduced. Their asymptotic distributions are derived, under the null hypothesis as well as under contiguous ARMA alternatives. The asymptotically maximin most powerful quadratic serial rank tests (for a given density) are then obtained. Because of their close similarity with the Box‐Pierce parametric test, we call them rank portmanteau tests. The asymptotic relative efficiencies (AREs) of the rank portmanteau tests with respect to one another and their AREs with respect to the corresponding Box‐Pierce and quadratic Spearman tests are derived.

Suggested Citation

  • Marc. Hallin & Jean‐François Ingenbleek & Madan L. Puri, 1987. "Linear And Quadratic Serial Rank Tests For Randomness Against Serial Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(4), pages 409-424, July.
    • Handle: RePEc:bla:jtsera:v:8:y:1987:i:4:p:409-424
    DOI: 10.1111/j.1467-9892.1987.tb00004.x
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    Cited by:

    1. Andy Kwan & Ah-Boon Sim & Yangru Wu, 2005. "On the size and power of normalized autocorrelation coefficients," Applied Financial Economics, Taylor & Francis Journals, vol. 15(1), pages 1-11.
    2. Nasri, Bouchra R., 2022. "Tests of serial dependence for multivariate time series with arbitrary distributions," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    3. Lanh Tran & Berlin Wu, 1993. "Order statistics for nonstationary time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(4), pages 665-686, December.
    4. M'hammed Kadri & Khalid Rifi, 2002. "Asymptotic Bound on the Characteristic Function of Signed Linear Serial Rank Statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(2), pages 391-403, June.
    5. Marc Hallin & Khalid Rifi, 1997. "A Berry-Esséen Theorem for Serial Rank Statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(4), pages 777-799, December.
    6. Husková, M., 2003. "Serial rank statistics for detection of changes," Statistics & Probability Letters, Elsevier, vol. 61(2), pages 199-213, January.

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