A Note on Bootstrap for Gupta’s Subset Selection Procedure

Abstract

This study introduces a method of selecting a subset of k populations containing the best when the populations are ranked in terms of the population means. It is assumed that the populations have an unknown location family of distribution functions. The proposed method involves estimating the constant in Gupta’s subset selection procedure by bootstrap. It is shown that estimating this constant amounts to estimating the distribution function of a certain function of random variables. The proposed bootstrap method is shown to be consistent and second-order correct in the sense that the accuracy of bootstrap approximation is better than that of the approximation based on limiting distribution. Results of a simulation study are given.

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References

  1. Babu, G.J. and Singh, K. (1983). Inference on means using the bootstrap. Ann. Stat.11, 999–1003.

    MathSciNet  Article  Google Scholar 

  2. Babu, G.J. and Singh, K. (1984). On one-term Edgeworth correction by Efron’s bootstrap. Sankya A46, 219–232.

    MathSciNet  MATH  Google Scholar 

  3. Bhattacharya, R.N. (1985). Some recent results on Cramer-Edgeworth expansions with applications. Multivariate Analysis Vol. VI (P.R. Krishnaiah, ed.)

  4. Bhattacharya, R.N. (1987). Some aspects of Edgeworth expansions in statistics and probability. Wiley, New York, Bhattacharya, R. (ed.), p. 157–171.

  5. Bhattacharya, R.N. (1990). Asymptotic expansions in statistics. Springer, Berlin, Denker, M. and Bhattacharya, R. N. (eds.), p. 11–66.

  6. Bhattacharya, R.N. and Ghosh, J. (1978). On the validity of the formal edgeworth expansion. Ann. Stat.6, 434–451.

    MathSciNet  Article  Google Scholar 

  7. Bhattacharya, R.N. and Rao, R.R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.

    Google Scholar 

  8. Bickel, P.J. and Freedman, D.A. (1981). Some asymptotic theory for the bootstrap. Ann. Stat.9, 1196–1217.

    MathSciNet  Article  Google Scholar 

  9. Chow, Y.S. and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales, 2nd edn. Springer, Berlin.

    Google Scholar 

  10. Cui, X. and Wilson, J. (2008). On the probability of correct selection for large k populations, with application to microarray data. Biom. J.50, 5, 870–83.

    MathSciNet  Article  Google Scholar 

  11. Dourleijn, C.J. and Driessen, S.G.A.J. (1993). Subset selection procedures for randomized designs. Biom. J.35, 3, 267–282.

    Article  Google Scholar 

  12. Dourleijn, J. (1995). Subset selection in plant breeding practice. Euphytica81, 2, 207–216.

    Article  Google Scholar 

  13. Dourleijn, J. (1996). A case study of subset selection in sugar beet breeding. Journal of Statistical Planning and Inference54, 3, 323–344.

    Article  Google Scholar 

  14. Dudewicz, E. and Koo, J.O. (1982). The Complete Categorized Guide to Statistical Selection and Ranking Procedures. Columbus, American Sciences Press.

    Google Scholar 

  15. Dudley, R.M. (1989). Real Analysis and Probability. Wadsworth and Brooks.

  16. Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Stat.7, 1, 1–26.

    MathSciNet  Article  Google Scholar 

  17. Gupta, S.S. (1956). On a decision rule for a problem in ranking means. Mimeo. Series No. 150. Institute of Statistics University of North Carolina, Chapel Hill.

    Google Scholar 

  18. Gupta, S.S. (1965). On some multiple decision (selection and ranking) rules. Technometrics7, 2, 225–245.

    Article  Google Scholar 

  19. Gupta, S.S. and Huang, D. (1974). A note on selecting a subset of normal populations with unequal sample sizes. Sankhya B36, 389–396.

    MathSciNet  MATH  Google Scholar 

  20. Gupta, S.S. and Huang, D. (1976). Subset selection procedures for the means and variances of normal populations: unequal sample sizes case. Sankhya B38, 112–128.

    MathSciNet  MATH  Google Scholar 

  21. Gupta, S.S. and McDonald, G.C. (1970). On some classes of selection procedures based on ranks. Cambridge University Press, London, Puri, M. L. (ed.), p. 491–514.

  22. Gupta, S.S. and Panchapakesan, S. (1979). Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations. Wiley, New York.

    Google Scholar 

  23. Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.

    Google Scholar 

  24. Helmers, R. (1991). On the edgeworth expansion and the bootstrap approximation for a studentized U-Statistic. Ann. Stat.19, 1, 470–484.

    MathSciNet  Article  Google Scholar 

  25. Herrendörfer, G. and Tuchscherer, A. (1996). Selection and breeding. Journal of Statistical Planning and Inference54, 3, 307–321.

    MathSciNet  Article  Google Scholar 

  26. Horrace, W.C. (2006). Selection procedures for economics. Appl. Econ. Q.52, 4, 1–18.

    MathSciNet  Google Scholar 

  27. Horrace, W.C., Marchand, J.T. and Smeeding, T. (2008). Ranking inequality: Applications of multivariate subset selection. J. Econ. Inequal.6, 1, 5–32.

    Article  Google Scholar 

  28. Kannan, P.K. and Sanchez, S.M. (1994). Competitive market structures: a subset selection analysis. Manag. Sci.40, 11, 1484–1499.

    Article  Google Scholar 

  29. Kim, S.H. and Nelson, B. (2005). Selecting the best system. Elsevier, Oxford, p. 501–534.

  30. Listing, J. and Rasch, D. (1996). Robustness of subset selection procedures. Journal of Statistical Planning and Inference54, 291–305.

    MathSciNet  Article  Google Scholar 

  31. Rizvi, M.H. and Sobel, M. (1967). Nonparametric procedures for selecting a subset containing the population with the largest α-quantile. Ann. Math. Stat.38, 1788–1803.

    MathSciNet  Article  Google Scholar 

  32. Shao, J. and Tu, D (1995). The Jackknife and Bootstrap. Springer, New York.

    Google Scholar 

  33. Singh, K. (1981). On asymptotic accuracy of Efron’s bootstrap. Ann. Stat.9, 1187–1195.

    MathSciNet  Article  Google Scholar 

  34. Swanepoel, J.W.H. (1983). Bootstrap selection procedures based on robust estimators. Communications in Statistics - Theory and Methods12, 18, 2059–2083.

    MathSciNet  Article  Google Scholar 

  35. Swanepoel, J.W.H. (1985). Bootstrap selection procedures based on robust estimators (with discussion). American Sciences Press, Columbus, Dudewicz, E. J. (ed.), p. 45–64.

  36. van der Vaart, A.W. (2000). Asymptotic Statistics. Cambridge University Press, Cambridge.

    Google Scholar 

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Acknowledgements

The main part of this study is done while author is a visiting professor of University of Tokyo. The author would like to thank Professor Hiroshi Kurata for his hospitality. The author also wish to thank Professor Satoshi Kuriki and two anonymous referees for valuable suggestions and comments.

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Correspondence to Jun-ichiro Fukuchi.

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Fukuchi, J. A Note on Bootstrap for Gupta’s Subset Selection Procedure. Sankhya A 82, 96–114 (2020). https://doi.org/10.1007/s13171-019-00163-6

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Keywords and phrases

  • Bootstrap
  • Selection problem
  • Subset selection approach
  • Second-order correctness
  • Edgeworth expansion

AMS (2000) subject classification

  • Primary 62G09
  • Secondary 62F07