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Asymptotic expansions for the power of adaptive rank tests in the one-sample problem

Part of the Lecture Notes in Mathematics book series (LNM,volume 821)

Abstract

In this paper we consider adaptive rank tests for the one-sample problem. Here adaptation means that the score function J of the rank test is estimated from the sample. We restrict attention to cases with a moderate degree of adaptation, in the sense that we require that the estimated J belongs to a one-parameter family J={Jr|rєI⊂R1}. For the power of adaptive rank tests of this type, we establish asymptotic expansions under contiguous location alternatives, for general estimators S of the parameter r. These expansions are used to compare, in terms of deficiencies, the performance of these adaptive rank tests to that of rank tests with fixed scores. Conditions on S and Jr are given under which the deficiency tends to a finite limit, which is obtained. It is verified that these conditions hold for a particular class of estimators which are related to the sample kurtosis. In this case explicit results are obtained.

This research was supported by the Netherlands Organization for the Advancement of of Pure Research (Z.W.O.).

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References

  1. Albers, W., Bickel, P. J. and van Zwet, W.R. (1976). Asymptotic expansions for the power of distributionfree tests in the one-sample problem. Ann. Statist. 4, 108–157.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Chung, K. L. (1951). The strong law of large numbers. Proc. 2nd Berkeley Symp. on Math. Statist. and Prob., 341–352.

    Google Scholar 

  3. Gastwirth, J. L. (1966). On robust procedures. J. Amer. Statist. Ass. 61, 929–948.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Gastwirth, J. L. (1970). On asymptotic relative efficiencies of a class of rank tests. J. Roy. Statist. Soc. Ser. B32, 227–232.

    MathSciNet  MATH  Google Scholar 

  5. Hájek, J. and Šidák, Z. (1967). Theory of rank tests. Academic Press, New York.

    MATH  Google Scholar 

  6. Hodges, J. L. Jr. and Lehmann, E.L. (1970). Deficiency. Ann. Math. Statist. 41, 783–801.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Hogg, R. V. (1967). Some observations on robust estimation. J. Amer. Statist. Ass. 62, 1179–1186.

    CrossRef  MathSciNet  Google Scholar 

  8. Hogg, R.V. (1972). More light on the kurtosis and related statistics. J. Amer. Statist. Ass. 67, 422–424.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Huber, P.J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Huber, P.J. (1972). Robust statistics: a review. Ann. Math. Statist. 43, 1041–1067.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Jaeckel, L.A. (1971). Some flexible estimates of location. Ann. Math. Statist. 42, 1540–1552.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Shapiro, S.S., Wilk, M.B. and Chen, H.J. (1968). A comparative study of various tests for normality. J. Amer. Statist. Ass. 63, 1343–1372.

    CrossRef  MathSciNet  Google Scholar 

  13. Stein, C. (1956). Efficient nonparametric testing and estimation. Proc. Third Berkeley Symp. Math. Statist. and Prob. I, 187–195.

    MathSciNet  MATH  Google Scholar 

  14. Stigler, S.M. (1969). Linear functions of order statistics. Ann. Math. Statist. 40, 770–788.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. van Zwet, W.R. (1964). Convex transformations of random variables. Math. Centre Tracts 7, Mathematisch Centrum, Amsterdam.

    MATH  Google Scholar 

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© 1980 Springer-Verlag

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Albers, W. (1980). Asymptotic expansions for the power of adaptive rank tests in the one-sample problem. In: Raoult, JP. (eds) Statistique non Paramétrique Asymptotique. Lecture Notes in Mathematics, vol 821. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097427

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  • DOI: https://doi.org/10.1007/BFb0097427

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10239-7

  • Online ISBN: 978-3-540-38318-5

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