Environmental and Ecological Statistics

, Volume 6, Issue 2, pp 105–118 | Cite as

Distribution-free ranked-set sample procedures for umbrella alternatives in the m-sample setting

  • Bradley A. Hartlaub
  • Douglas A. Wolfe
Article

Abstract

In settings where measurements are costly and/or difficult to obtain but ranking of the potential sample data is relatively easy and reliable, the use of statistical methods based on a ranked-set sampling approach can lead to substantial improvement over analogous methods associated with simple random samples. Previous nonparametric work in this area has been concentrated almost exclusively on the one- and two-sample location problems. In this paper we develop ranked-set sample procedures for the m-sample location setting where the treatment effect parameters follow a restricted umbrella pattern. Distribution-free testing procedures are developed for both the case where the peak of the umbrella is known and for the case where it is unknown. Small sample and asymptotic null distribution properties are provided for the peak-known test statistic.

distribution-free m-sample setting ranked-set sampling umbrella alternatives 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bohn, L.L. (1994) A Ranked-Set Sample Signed-Rank Statistic. Department of Statistics Technical Report Number 426. The University of Florida, Gainesville, FL.Google Scholar
  2. Bohn, L.L. (1996) A review of nonparametric ranked-set sampling methodology. Communications in Statistics, Theory and Methods, 25(11), 2675-85.Google Scholar
  3. Bohn, L.L. and Wolfe, D.A. (1992) Nonparametric two-sample procedures for ranked-set samples data. Journal of the American Statistical Association, 87, 552-61.Google Scholar
  4. Bohn, L.L. and Wolfe, D.A. (1994) The effect of imperfect judgment rankings on properties of procedures based on the ranked-set samples analog of the Mann-Whitney-Wilcoxon statistic. Journal of the American Statistical Association, 89, 168-76.Google Scholar
  5. Chen, Y.I. and Wolfe, D.A. (1990) A study of distribution-free tests for umbrella alternatives. Biometrical Journal, 32, 47-57.Google Scholar
  6. Dell, T.R. and Clutter, J.L. (1972) Ranked-set sampling theory with order statistics background. Biometrics, 28, 545-55.Google Scholar
  7. Hartlaub, B.A. and Wolfe, D.A. (1998) Extended Tables of Null Critical Values for the Bohn-Wolfe Ranked-Set Sample Analogue of the Mann-Whitney-Wilcoxon Two-Sample Statistic. Department of Statistics Technical Report Number 625, Ohio State University, Columbus, OH.Google Scholar
  8. Hettmansperger, T.P. (1995) The ranked-set sample sign test. Journal of Nonparametric Statistics, 4, 263-70.Google Scholar
  9. Kaur, A., Patil, G.P., Sinha, A.K., and Taillie, C. (1995) Unequal Allocation Issues in Ranked-Set Sampling. Center for Statistical Ecology and Environmental Statistics Technical Report Number 95-0703. The Pennsylvania State University, University Park, PA.Google Scholar
  10. Kruskal, W.H. and Wallis, W.A. (1952) Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47, 583-621.Google Scholar
  11. Kvam, P.H. and Samaniego, F.J. (1993) On the inadmissibility of empirical averages as estimators in ranked-set sampling. Journal of Statistical Planning and Inference, 36, 39-55.Google Scholar
  12. Kvam, P.H. and Samaniego, F.J. (1994) Nonparametric maximum likelihood estimation based on rankedset samples. Journal of the American Statistical Association, 89, 526-37.Google Scholar
  13. Mack, G.A. and Wolfe, D.A. (1981) K-sample rank tests for umbrella alternatives. Journal of the American Statistical Association, 76, 175-81.Google Scholar
  14. McIntyre, G.A. (1952) A method of unbiased selective sampling, using ranked sets. Australian Journal of Agricultural Research, 3, 385-90.Google Scholar
  15. Randles, R.H. and Wolfe, D.A. (1979) Introduction to the Theory of Nonparametric Statistics, John Wiley, New York.Google Scholar
  16. Stokes, S.L. and Sager, T.W. (1988) Characterizations of a ranked-set sample with application to estimating distribution functions. Journal of the American Statistical Association, 83, 374-81.Google Scholar
  17. Takahasi, K. and Wakimoto, K. (1968) On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics, 20, 1-31.Google Scholar
  18. Terpstra, T.J. (1952) The asymptotic normality and consistency of Kendall's test against trend, when ties are present in one ranking. Indagationes Mathematicae, 14, 327-33.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Bradley A. Hartlaub
    • 1
  • Douglas A. Wolfe
    • 2
  1. 1.Department of MathematicsKenyon CollegeGambier
  2. 2.Department of StatisticsThe Ohio State UniversityColumbus

Personalised recommendations