Advertisement

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

  • Willem Albers
Conference paper
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [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. [3]
    Gastwirth, J. L. (1966). On robust procedures. J. Amer. Statist. Ass. 61, 929–948.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Gastwirth, J. L. (1970). On asymptotic relative efficiencies of a class of rank tests. J. Roy. Statist. Soc. Ser. B32, 227–232.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Hájek, J. and Šidák, Z. (1967). Theory of rank tests. Academic Press, New York.zbMATHGoogle Scholar
  6. [6]
    Hodges, J. L. Jr. and Lehmann, E.L. (1970). Deficiency. Ann. Math. Statist. 41, 783–801.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Hogg, R. V. (1967). Some observations on robust estimation. J. Amer. Statist. Ass. 62, 1179–1186.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Hogg, R.V. (1972). More light on the kurtosis and related statistics. J. Amer. Statist. Ass. 67, 422–424.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Huber, P.J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Huber, P.J. (1972). Robust statistics: a review. Ann. Math. Statist. 43, 1041–1067.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Jaeckel, L.A. (1971). Some flexible estimates of location. Ann. Math. Statist. 42, 1540–1552.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [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.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Stein, C. (1956). Efficient nonparametric testing and estimation. Proc. Third Berkeley Symp. Math. Statist. and Prob. I, 187–195.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Stigler, S.M. (1969). Linear functions of order statistics. Ann. Math. Statist. 40, 770–788.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    van Zwet, W.R. (1964). Convex transformations of random variables. Math. Centre Tracts 7, Mathematisch Centrum, Amsterdam.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Willem Albers
    • 1
  1. 1.Technological University TwenteEnschede

Personalised recommendations