Sankhya A

pp 1–19 | Cite as

A Note on Bootstrap for Gupta’s Subset Selection Procedure

  • Jun-ichiro FukuchiEmail author


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.

Keywords and phrases

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

AMS (2000) subject classification

Primary 62G09 Secondary 62F07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


  1. Babu, G.J. and Singh, K. (1983). Inference on means using the bootstrap. Ann. Stat. 11, 999–1003.MathSciNetzbMATHGoogle Scholar
  2. Babu, G.J. and Singh, K. (1984). On one-term Edgeworth correction by Efron’s bootstrap. Sankya A 46, 219–232.MathSciNetzbMATHGoogle Scholar
  3. Bhattacharya, R.N. (1985). Some recent results on Cramer-Edgeworth expansions with applications. Multivariate Analysis Vol. VI (P.R. Krishnaiah, ed.)Google Scholar
  4. Bhattacharya, R.N. (1987). Some aspects of Edgeworth expansions in statistics and probability. Wiley, New York, Bhattacharya, R. (ed.), p. 157–171.Google Scholar
  5. Bhattacharya, R.N. (1990). Asymptotic expansions in statistics. Springer, Berlin, Denker, M. and Bhattacharya, R. N. (eds.), p. 11–66.Google Scholar
  6. Bhattacharya, R.N. and Ghosh, J. (1978). On the validity of the formal edgeworth expansion. Ann. Stat. 6, 434–451.MathSciNetzbMATHGoogle Scholar
  7. Bhattacharya, R.N. and Rao, R.R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.zbMATHGoogle Scholar
  8. Bickel, P.J. and Freedman, D.A. (1981). Some asymptotic theory for the bootstrap. Ann. Stat. 9, 1196–1217.MathSciNetzbMATHGoogle Scholar
  9. Chow, Y.S. and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales, 2nd edn. Springer, Berlin.zbMATHGoogle 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.MathSciNetGoogle Scholar
  11. Dourleijn, C.J. and Driessen, S.G.A.J. (1993). Subset selection procedures for randomized designs. Biom. J. 35, 3, 267–282.zbMATHGoogle Scholar
  12. Dourleijn, J. (1995). Subset selection in plant breeding practice. Euphytica 81, 2, 207–216.Google Scholar
  13. Dourleijn, J. (1996). A case study of subset selection in sugar beet breeding. Journal of Statistical Planning and Inference 54, 3, 323–344.zbMATHGoogle Scholar
  14. Dudewicz, E. and Koo, J.O. (1982). The Complete Categorized Guide to Statistical Selection and Ranking Procedures. Columbus, American Sciences Press.zbMATHGoogle Scholar
  15. Dudley, R.M. (1989). Real Analysis and Probability. Wadsworth and Brooks.Google Scholar
  16. Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Stat. 7, 1, 1–26.MathSciNetzbMATHGoogle 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. Technometrics 7, 2, 225–245.zbMATHGoogle Scholar
  19. Gupta, S.S. and Huang, D. (1974). A note on selecting a subset of normal populations with unequal sample sizes. Sankhya B 36, 389–396.MathSciNetzbMATHGoogle 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 B 38, 112–128.MathSciNetzbMATHGoogle 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.Google Scholar
  22. Gupta, S.S. and Panchapakesan, S. (1979). Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations. Wiley, New York.zbMATHGoogle Scholar
  23. Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.zbMATHGoogle Scholar
  24. Helmers, R. (1991). On the edgeworth expansion and the bootstrap approximation for a studentized U-Statistic. Ann. Stat. 19, 1, 470–484.MathSciNetzbMATHGoogle Scholar
  25. Herrendörfer, G. and Tuchscherer, A. (1996). Selection and breeding. Journal of Statistical Planning and Inference 54, 3, 307–321.MathSciNetzbMATHGoogle Scholar
  26. Horrace, W.C. (2006). Selection procedures for economics. Appl. Econ. Q. 52, 4, 1–18.MathSciNetGoogle 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.Google Scholar
  28. Kannan, P.K. and Sanchez, S.M. (1994). Competitive market structures: a subset selection analysis. Manag. Sci. 40, 11, 1484–1499.zbMATHGoogle Scholar
  29. Kim, S.H. and Nelson, B. (2005). Selecting the best system. Elsevier, Oxford, p. 501–534.Google Scholar
  30. Listing, J. and Rasch, D. (1996). Robustness of subset selection procedures. Journal of Statistical Planning and Inference 54, 291–305.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  32. Shao, J. and Tu, D (1995). The Jackknife and Bootstrap. Springer, New York.zbMATHGoogle Scholar
  33. Singh, K. (1981). On asymptotic accuracy of Efron’s bootstrap. Ann. Stat. 9, 1187–1195.MathSciNetzbMATHGoogle Scholar
  34. Swanepoel, J.W.H. (1983). Bootstrap selection procedures based on robust estimators. Communications in Statistics - Theory and Methods 12, 18, 2059–2083.MathSciNetzbMATHGoogle 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.Google Scholar
  36. van der Vaart, A.W. (2000). Asymptotic Statistics. Cambridge University Press, Cambridge.Google Scholar

Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Faculty of EconomicsGakushuin UniversityTokyoJapan

Personalised recommendations