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Asymptotic distribution of rank statistics under dependencies with multivariate application

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  • Thompson, G. L.

Abstract

By modifying the method of projection, the results of Hajek and Huskova are extended to show the asymptotic normality of signed and linear rank statistics under general alternatives for dependent random variables that can be expressed as independent vectors of fixed equal length. The score function is twice differentiable; the regression constants are arbitrary; and the distribution functions are continuous, but arbitrary. As an application, a rank transform statistic is proposed for the one-sample multivariate location model. The ranks of the absolute values of the observations are calculated without regard to component membership, and the scored ranks are substituted in place of the observed values. The limiting distribution of the proposed test statistic is shown to be [chi]2 divided by the degrees of freedom under the null hypothesis, and noncentral [chi]2 divided by the degrees of freedom under the sequence of Pitman alternatives.

Suggested Citation

  • Thompson, G. L., 1990. "Asymptotic distribution of rank statistics under dependencies with multivariate application," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 183-211, May.
    • Handle: RePEc:eee:jmvana:v:33:y:1990:i:2:p:183-211
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    Citations

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    Cited by:

    1. Bathke, Arne C. & Harrar, Solomon W. & Madden, Laurence V., 2008. "How to compare small multivariate samples using nonparametric tests," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4951-4965, July.
    2. Fan, Chunpeng & Zhang, Donghui, 2014. "Wald-type rank tests: A GEE approach," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 1-16.
    3. Harrar, Solomon W. & Feyasa, Merga B. & Wencheko, Eshetu, 2020. "Nonparametric procedures for partially paired data in two groups," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    4. Liu, Chunxu & Bathke, Arne C. & Harrar, Solomon W., 2011. "A nonparametric version of Wilks' lambda--Asymptotic results and small sample approximations," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1502-1506, October.
    5. Harrar, Solomon W. & Bathke, Arne C., 2008. "Nonparametric methods for unbalanced multivariate data and many factor levels," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1635-1664, September.
    6. Edgar Brunner & Madan Puri, 2001. "Nonparametric methods in factorial designs," Statistical Papers, Springer, vol. 42(1), pages 1-52, January.
    7. Sebastian Domhof & Edgar Brunner & D. Wayne Osgood, 2002. "Rank Procedures for Repeated Measures with Missing Values," Sociological Methods & Research, , vol. 30(3), pages 367-393, February.
    8. Harrar, Solomon W. & Kong, Xiaoli, 2022. "Recent developments in high-dimensional inference for multivariate data: Parametric, semiparametric and nonparametric approaches," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    9. Wang, Haiyan & Akritas, Michael G., 2010. "Rank test for heteroscedastic functional data," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1791-1805, September.
    10. Brunner, Edgar & Munzel, Ulrich & Puri, Madan L., 1999. "Rank-Score Tests in Factorial Designs with Repeated Measures," Journal of Multivariate Analysis, Elsevier, vol. 70(2), pages 286-317, August.

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