Selecting the Best Population Using a Test for Equality Based on Minimal Wilcoxon Rank-sum Precedence Statistic

  • Hon Keung Tony NgEmail author
  • N. Balakrishnan
  • S. Panchapakesan


In this paper, we first give an overview of the precedence-type test procedures. Then we propose a nonparametric test based on early failures for the equality of two life-time distributions against two alternatives concerning the best population. This procedure utilizes the minimal Wilcoxon rank-sum precedence statistic (Ng and Balakrishnan, 2002, 2004) which can determine the difference between populations based on early (100q%) failures. Hence, this procedure can be useful in life-testing experiments in biological as well as industrial settings. After proposing the test procedure, we derive the exact null distribution of the test statistic in the two-sample case with equal or unequal sample sizes. We also present the exact probability of correct selection under the Lehmann alternative. Then, we generalize the test procedure to the k-sample situation. Critical values for some sample sizes are presented. Next, we examine the performance of this test procedure under a location-shift alternative through Monte Carlo simulations. Two examples are presented to illustrate our test procedure with selecting the best population as an objective.


Two-sample problem k-sample problem Precedence statistics Life-testing Lehmann alternative Monte Carlo simulations Probability of correct selection Wilcoxon rank-sum statistic 

AMS 2000 Subject Classification

62G10 62N05 


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  1. N. Balakrishnan and R. Frattina, “Precedence test and maximal precedence test.” In N. Limnios and M. Nikulin (eds.), Recent Advances in Reliability Theory: Methodology, Practice, and Inference, pp. 355–378, Birkhäuser: Boston, MA, 2000.Google Scholar
  2. N. Balakrishnan and M. V. Koutras, Runs and Scans with Applications, Wiley: New York, 2002.zbMATHGoogle Scholar
  3. N. Balakrishnan and H. K. T. Ng, “A general maximal precedence test.” In Y. Hayakawa, T. Irony, and M. Xie (eds.), System and Bayesian Reliability – Essays in Honor of Prof. Richard E. Barlow on his 70th Birthday, pp. 105–122, World Scientific: Singapore, 2001.Google Scholar
  4. N. Balakrishnan and H. K. T. Ng, Precedence-Type Tests and Applications, Wiley: Hoboken, NJ, 2006.zbMATHCrossRefGoogle Scholar
  5. N. Balakrishnan, H. K. T. Ng, and S. Panchapakesan, “A nonparametric procedure based on early failures for selecting the best population using a test for equality,” Journal of Statistical Planning and Inference vol. 136 pp. 2087–2111, 2006.Google Scholar
  6. S. Chakraborti and P. van der Laan, “Precedence tests and confidence bounds for complete data: an overview and some results,” The Statistician vol. 45 pp. 351–369, 1996.CrossRefGoogle Scholar
  7. S. Chakraborti and P. van der Laan, “An overview of precedence-type tests for censored data,” Biometrical Journal vol. 39 pp. 99–116, 1997.zbMATHGoogle Scholar
  8. R. B. Davies, “Rank tests for Lehmann alternative,” Journal of the American Statistical Association. vol. 66 pp. 879–883, 1971.zbMATHCrossRefGoogle Scholar
  9. J. Eilbott and J. Nadler, “On precedence life testing,” Technometrics vol. 7 pp. 359–377, 1965.CrossRefMathSciNetGoogle Scholar
  10. J. D. Gibbons and S. Chakraborti, Nonparametric Statistical Inference, 3rd edn., Marcel Dekker: New York, 1992.zbMATHGoogle Scholar
  11. M. Hollander and D. A. Wolfe, Nonparametric Statistical Methods, 2nd edn, Wiley: New York, 1999.zbMATHGoogle Scholar
  12. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 1, 2nd edn, Wiley: New York, 1994.zbMATHGoogle Scholar
  13. E. L. Lehmann, “The power of rank tests,” Annals of Mathematical Statistics vol. 24 pp. 23–42, 1953.MathSciNetGoogle Scholar
  14. E. L. Lehmann, Nonparametrics: Statistical Methods Based on Ranks, 2nd edn., McGraw-Hill: New York, 1998.Google Scholar
  15. C. H. Lin and S. Sukhatme, “On the choice of precedence tests,” Communications in Statistics– Theory and Methods vol. 21 pp. 2949–2968, 1992.zbMATHMathSciNetGoogle Scholar
  16. L. S. Nelson, “Tables of a precedence life test,” Technometrics vol. 5 pp. 491–499, 1963.zbMATHCrossRefGoogle Scholar
  17. L. S. Nelson “Precedence life test.” In S. Kotz and N. L. Johnson(eds.), Encyclopedia of Statistical Sciences vol. 7, pp. 134–136, Wiley: New York, 1986.Google Scholar
  18. L. S. Nelson, “Tests on early failures – the precedence life test,” Journal of Quality Technology vol. 25 pp. 140–143, 1993.Google Scholar
  19. W. Nelson, Applied Life Data Analysis, Wiley: New York, 1982.zbMATHCrossRefGoogle Scholar
  20. H. K. T. Ng and N. Balakrishnan, “Wilcoxon-type rank-sum precedence tests: Large-sample approximation and evaluation,” Applied Stochastic Models in Business and Industry vol. 18 pp. 271–286, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  21. H. K. T. Ng and N. Balakrishnan, “Wilcoxon-type rank-sum precedence tests,” Australia and New Zealand Journal of Statistics vol. 46 pp. 631–648, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  22. H. K. T. Ng and N. Balakrishnan, “Precedence Testing,” In S. Kotz, N. Balakrishnan, C. B. Read, and B. Vidakovic (eds.), Encyclopedia of Statistical Sciences, 2nd edn. vol. 9 pp. 6317–6323, Wiley: Hoboken, NJ, 2006.Google Scholar
  23. R. A. Shorack, “On the power of precedence life tests,” Technometrics vol. 9 pp. 154–158, 1967.zbMATHCrossRefMathSciNetGoogle Scholar
  24. P. van der Laan and S. Chakraborti, “Precedence tests and Lehmann alternatives,” Statistical Papers vol. 42 pp. 301–312, 2001.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Hon Keung Tony Ng
    • 1
    Email author
  • N. Balakrishnan
    • 2
  • S. Panchapakesan
    • 3
  1. 1.Department of Statistical ScienceSouthern Methodist UniversityDallasUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  3. 3.Department of MathematicsSouthern Illinois University at CarbondaleCarbondaleUSA

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