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
Babu, G.J. and Singh, K. (1983). Inference on means using the bootstrap. Ann. Stat.11, 999–1003.
Babu, G.J. and Singh, K. (1984). On one-term Edgeworth correction by Efron’s bootstrap. Sankya A46, 219–232.
Bhattacharya, R.N. (1985). Some recent results on Cramer-Edgeworth expansions with applications. Multivariate Analysis Vol. VI (P.R. Krishnaiah, ed.)
Bhattacharya, R.N. (1987). Some aspects of Edgeworth expansions in statistics and probability. Wiley, New York, Bhattacharya, R. (ed.), p. 157–171.
Bhattacharya, R.N. (1990). Asymptotic expansions in statistics. Springer, Berlin, Denker, M. and Bhattacharya, R. N. (eds.), p. 11–66.
Bhattacharya, R.N. and Ghosh, J. (1978). On the validity of the formal edgeworth expansion. Ann. Stat.6, 434–451.
Bhattacharya, R.N. and Rao, R.R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.
Bickel, P.J. and Freedman, D.A. (1981). Some asymptotic theory for the bootstrap. Ann. Stat.9, 1196–1217.
Chow, Y.S. and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales, 2nd edn. Springer, Berlin.
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.
Dourleijn, C.J. and Driessen, S.G.A.J. (1993). Subset selection procedures for randomized designs. Biom. J.35, 3, 267–282.
Dourleijn, J. (1995). Subset selection in plant breeding practice. Euphytica81, 2, 207–216.
Dourleijn, J. (1996). A case study of subset selection in sugar beet breeding. Journal of Statistical Planning and Inference54, 3, 323–344.
Dudewicz, E. and Koo, J.O. (1982). The Complete Categorized Guide to Statistical Selection and Ranking Procedures. Columbus, American Sciences Press.
Dudley, R.M. (1989). Real Analysis and Probability. Wadsworth and Brooks.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Stat.7, 1, 1–26.
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.
Gupta, S.S. (1965). On some multiple decision (selection and ranking) rules. Technometrics7, 2, 225–245.
Gupta, S.S. and Huang, D. (1974). A note on selecting a subset of normal populations with unequal sample sizes. Sankhya B36, 389–396.
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.
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.
Gupta, S.S. and Panchapakesan, S. (1979). Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations. Wiley, New York.
Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
Helmers, R. (1991). On the edgeworth expansion and the bootstrap approximation for a studentized U-Statistic. Ann. Stat.19, 1, 470–484.
Herrendörfer, G. and Tuchscherer, A. (1996). Selection and breeding. Journal of Statistical Planning and Inference54, 3, 307–321.
Horrace, W.C. (2006). Selection procedures for economics. Appl. Econ. Q.52, 4, 1–18.
Horrace, W.C., Marchand, J.T. and Smeeding, T. (2008). Ranking inequality: Applications of multivariate subset selection. J. Econ. Inequal.6, 1, 5–32.
Kannan, P.K. and Sanchez, S.M. (1994). Competitive market structures: a subset selection analysis. Manag. Sci.40, 11, 1484–1499.
Kim, S.H. and Nelson, B. (2005). Selecting the best system. Elsevier, Oxford, p. 501–534.
Listing, J. and Rasch, D. (1996). Robustness of subset selection procedures. Journal of Statistical Planning and Inference54, 291–305.
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.
Shao, J. and Tu, D (1995). The Jackknife and Bootstrap. Springer, New York.
Singh, K. (1981). On asymptotic accuracy of Efron’s bootstrap. Ann. Stat.9, 1187–1195.
Swanepoel, J.W.H. (1983). Bootstrap selection procedures based on robust estimators. Communications in Statistics - Theory and Methods12, 18, 2059–2083.
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.
van der Vaart, A.W. (2000). Asymptotic Statistics. Cambridge University Press, Cambridge.
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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Fukuchi, Ji. 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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DOI: https://doi.org/10.1007/s13171-019-00163-6