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Subset selection procedures: Review and assessment. (English) Zbl 0633.62024

Ranking and selection problems have customarily been treated using two different approaches, namely, the indifference-zone approach and the subset-selection approach. This paper reviews the development of the subset-selection (also known as “Gupta’s formulation”) over the years 1950-1980; the indifference-zone approach is also discussed.
The goal of the subset-selection approach is to select a random size subset of the k populations so that the best is included in the selected subset with a guaranteed minimum probability. In the indifference-zone approach, one wants to select one of the k populations so that the probability of a correct selection achieves certain probability requirement. The authors of this paper divide the developments in the years 1950-1980 into three periods: the years of early development, the years of growth, and the years of further strides. They also access the impact of these developments on users in the last part of the paper. The paper concludes with more than 15 pages of references.
In the first decade 1955-1965, subset-selection rules predictably centered around well-known parametric families of distribution, namely, the normal, binomial, and gamma. Besides the problem of selecting the best of k given populations, procedures were proposed and investigated in this period also for the problem of selecting a subset containing all the populations that are better than a control, and that of partitioning a set of populations with respect to a control.
The decade 1965-1975 witnessed significant growth of ranking and selection theory in general and of the subset-selection approach in particular. The main developments during this period were categorized by the authors into (1) unified theory; (2) generalizations and modifications of the basic formulations; (3) decision-theoretic formulation; Bayes and empirical Bayes procedures; (4) multivariate normal and multinomial selection; (5) conditional selection; (6) nonparametric procedures; (7) selection from restricted families; (8) sequential procedures; (9) other developments.
The authors view the developments of the decision-theoretic approach to subset-selection the major contributions made during the period 1975- 1980. Bayes procedures, minimax rules, construction of optimal selection procedures and essentially complete classes, locally optimal subset- selection, modified goal and a complete ranking formulation entropy-based selection and other developments in this period are carefully reviewed in this paper.
The last part of the paper discusses the development of the subset- selection approach in 1980-1984. It also gives assessment to the entire field of ranking and selection. This part of the paper is important since: (1) it contains a complete review of the recent developments after the publications of the books by J. D. Gibbons, I. Olkin and M. Sobel, Selecting and ordering populations: A new statistical methodology (1977; Zbl 0464.62022), and by the present authors, Multiple decision procedures: Theory and methodology of selecting and ranking populations (1979; Zbl 0516.62030);
(2) it provides possible directions to the researchers in this area and states some open problems in ranking and selection. Thus this part (and perhaps the entire paper) should be used as a supplement to either of the above-mentioned two books.
Reviewer: Chen Pinyuen

MSC:

62F07 Statistical ranking and selection procedures
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[1] Alam K., Annals of the Institute of Statistical Mathematics 22 pp 127– (1970) · Zbl 0222.62028
[2] Alam K., Annals of the Institute of Statistical Mathematics 18 pp 307– (1966) · Zbl 0148.13804
[3] Amer P. D., Communications in Statistics-Theory and Methods 9 pp 1461– (1980) · Zbl 0444.62033
[4] Arvesen J. N., Journal of the American Statistical Association 70 pp 166– (1975)
[5] Bahadur R. R., Annals of Mathematical Statistics 21 pp 362– (1950) · Zbl 0038.29702
[6] Bahadur R. R., Annals of Mathematical statistics 21 pp 469– (1950) · Zbl 0040.07501
[7] Barlow R. E., Annals of Mathematical Statistics 40 pp 905– (1969) · Zbl 0181.45902
[8] Barlow R. E., Annals Of Mathematical Statistics 40 pp 918– (1969) · Zbl 0181.45903
[9] Barr D. R., Journal of the American Statistical Association 61 pp 640– (1966)
[10] Barr D. R., Trabajos de Estadistica 17 pp 15– (1966)
[11] Barron A. M., A Class of Sequential Multiple Decision Procedures (1968)
[12] Barron A. M., Operations Research-Verfahren pp 11– (1972)
[13] Bechhofer R. E., Annals of Mathematical Statistics 25 pp 16– (1954) · Zbl 0055.13003
[14] Bechhofer R. E., Sequential Identification and Ranking Procedures (with special reference to Koopman-Darmois populations) (1968) · Zbl 0208.44601
[15] Becker W. A., Poultry Science 40 pp 1507– (1961)
[16] Becker W. A., Poultry Science 41 pp 1437– (1963)
[17] Berger R. L., Annals of Statistics 7 pp 1333– (1979) · Zbl 0418.62022
[18] Berger R. L., Scandinavian Journal of Statistics 7 pp 21– (1980)
[19] Bhattacharya P. K., Calcutta Statistical Association Bulletin 7 pp 1– (1956)
[20] Bickel P. J., statistical Decision Theory and Related Topics-II pp 37– (1977)
[21] Bickel P. J., Statistics and Probability: Essays in Honor of C. R. Rao pp 109– (1982)
[22] Bjørnstad J. F., Technometrics 22 pp 617– (1980)
[23] Bjørnstad J. F., Communications in Statistics - Theory and Methods 10 pp 2411– (1981) · Zbl 0484.62041
[24] Bjørnstad J. F., Annals of Statistics 9 pp 777– (1981) · Zbl 0475.62016
[25] Bjørnstad J. F., A Festschrift for Erich L. Lehmann pp 62– (1983)
[26] Blum J. R., Annals of Mathematical Statistics 38 pp 1671– (1967) · Zbl 0173.20404
[27] Blumenthal S., Sankhyā 31 pp 37– (1969)
[28] Broström G., Communications in Statistics - Theory and Methods 10 pp 203– (1981)
[29] Büringer H., Nonparametric Sequential Selection Procedures (1980) · Zbl 0462.62023
[30] Carroll R. J., Some Contributions to the Theory of Parametric and Nonparametric Sequential Ranking and Selection (1974)
[31] Carroll R. J., Communications in Statistics 4 pp 987– (1975)
[32] Chapman D. G., Annals of the Institute of Statistical Mathematics 4 pp 45– (1952) · Zbl 0049.22003
[33] Chattopadhyay A. K., Communications in Statistics - Theory and Methods 10 pp 31– (1981) · Zbl 0454.62024
[34] Chen H. J., Sankhyā 38 pp 249– (1976)
[35] Chernoff H., Statistical Decision Theory and Related Topics-II pp 93– (1977)
[36] Chew V., Comparisons Among Treatment Means in an Analysis of Variance (1977)
[37] Chew V., Proceedings of Florida State Horticultural Society 90 pp 214– (1977)
[38] Chotai J., Communications in Statistics - Theory and Methods 9 pp 1147– (1980) · Zbl 0439.62019
[39] Chotai J., Selection and Ranking Procedures Based on Likelihood Ratios (1980) · Zbl 0439.62019
[40] Cochran W. C., Experimental Designs (1950)
[41] Cohen D. S., A Two-Sample Decision Procedure for Ranking Means of Normal populations with a Common Known Variance (1959)
[42] Deely J. J., Multiple Decision Procedures from an Empirical Bayes Approach (1965)
[43] Deely J. J., Sankhyā 30 pp 37– (1968)
[44] Desu M. M., Annals of Mathematical Statistics 41 pp 1596– (1970) · Zbl 0226.62057
[45] Desu M. M., Biometrika 55 pp 401– (1968)
[46] Deverman J. N., Annals of Mathematical Statistics 40 pp 1870– (1969)
[47] Dudewicz E. J., Introduction to statistics and Probability (1976) · Zbl 0358.62001
[48] Dudewicz E. J., The Frontiers of Modern Statistical Inference Procedures 10 (1985) · Zbl 0564.00019
[49] Dudewicz E. J., A new subset selection theory for all best populations and its applications (1983)
[50] Dudewicz E. J., Sankhyā 37 pp 28– (1975)
[51] Dudewicz E. J., The Complete Categorized Guide to Statistical Selection and Ranking Procedures 6 (1982) · Zbl 0509.62021
[52] Edwards H. P., The American Statistician 38 pp 158– (1984)
[53] Edwards H. P., The Frontiers of Modern Statistical Inference Procedures 10 (1985)
[54] Fabian V., Annals of Mathematical Statistics 33 pp 248– (1962) · Zbl 0114.10402
[55] Frischtak R. M., Statistical Multiple-Decision Procedures for Some Multivariate Selection Problems (1973)
[56] Ghosh M., Annals of Statistics 1 pp 773– (1973) · Zbl 0277.62039
[57] Gibbons J. D., Selecting and Ordering Populations: A New Statistical Methodology (1977) · Zbl 0464.62022
[58] Gnanadesikan M., Some Selection and Ranking Procedures for Multivariate Normal Populations (1966)
[59] Gnanadesikan M., Technometrics 12 pp 103– (1970)
[60] Goel P. K., Annals of Statistics 5 pp 969– (1977) · Zbl 0368.62013
[61] Gupta S. S., On a decision rule for a problem in ranking means (1956) · Zbl 0073.35901
[62] Gupta S. S., Annals of the Institute of Statistical Mathematics 14 pp 199– (1963) · Zbl 0128.13201
[63] Gupta S. S., Technometrics 7 pp 225– (1965)
[64] Gupta S. S., Multivariate Analysis pp 457– (1966)
[65] Gupta S. S., Statistical Decision Theory and Related Topics-III 2 (1982)
[66] Gupta S. S., Sankhyā 43 pp 291– (1981)
[67] Gupta S. S., Journal of Statistical Planning and Inference 8 pp 87– (1983) · Zbl 0536.62010
[68] GuptaS. S. and HsuJ. C. (1977). On the monotonicity of Bayes subset selection procedures.Proceedings of the 41st Session of the International Statistical Institute, Vol. 47, Book 4, 208–211.
[69] Gupta S. S., Communications in Statistics - Simulation and Computation 7 pp 561– (1978) · Zbl 0393.62004
[70] Gupta S. S., A computer package for ranking, selection, and multiple comparisons with the best (1984)
[71] Gupta S. S., Bulletin of the Institute of Mathematics, Academia Sinica 2 pp 377– (1974)
[72] Gupta S. S., Applied Statistics pp 97– (1975)
[73] Gupta S. S., Statistical Inference and Related Topics pp 101– (1975)
[74] Gupta S. S., Sankhyā 38 pp 112– (1976)
[75] Gupta S. S., Sankhyā 38 pp 153– (1976)
[76] Gupta S. S., Statistical Decision Theory and Related Topics-II pp 139– (1977)
[77] Gupta S. S., Annals of Statistics 8 pp 1164– (1980) · Zbl 0459.62019
[78] Gupta S. S., Journal of Statistical Planning and Inference 4 pp 115– (1980) · Zbl 0443.62016
[79] Gupta S. S., Multiple Decision Theory: Recent Developments 6 (1981) · Zbl 0471.62026
[80] Gupta S. S., Design of Experiments: Ranking and Selection pp 67– (1984)
[81] Gupta S. S., Essays in Probability and Statistics pp 501– (1976)
[82] Gupta S. S., Optimizing Methods in Statistics pp 251– (1979)
[83] Gupta S. S., Sankhyā 36 pp 389– (1974)
[84] Gupta S. S., Sankhyā 43 pp 245– (1981)
[85] Gupta S. S., On isotonic selection rules for binomial populations better than a standard (1982)
[86] Gupta S. S., Recent Developments in Statistical Inference and Data Analysis pp 55– (1980)
[87] Gupta S. S., Communications in Statistics - Theory and Methods 10 pp 1043– (1981) · Zbl 0484.62040
[88] Gupta S. S., Design of Experiments: Ranking and Selection pp 77– (1984)
[89] Gupta S. S., Decision Information pp 277– (1979)
[90] Gupta S. S., On Bayes and empirical Bayes rule for selecting good populations (1983) · Zbl 0536.62010
[91] Gupta S. S., Isotonic procedures for selecting populations better than a standard: two-parameter exponential distributions (1983)
[92] Gupta S. S., Annals of the Institute of Statistical Mathematics 31 pp 235– (1979) · Zbl 0451.62021
[93] Gupta S. S., Nonparametric Techniques in Statistical Inference pp 491– (1970)
[94] Gupta S. S., Colloquia Mathematica Societatis János Bolyai, 32: Nonparametric Statistical Inference pp 361– (1982)
[95] Gupta S. S., Sankhyā 43 pp 1– (1981)
[96] Gupta S. S., Statistics and Probability: Essays in Honor of C. R. Rao pp 295– (1982)
[97] Gupta S. S., Sankhyā 46 pp 123– (1984)
[98] Gupta S. S., Annals Of Statistics 12 pp 336– (1984) · Zbl 0539.62035
[99] Gupta S. S., Statistical Decision Theory and Related Topics-II (1977)
[100] Gupta S. S., Sankhyā 29 pp 1– (1967)
[101] Gupta S. S., Statistical Decision Theory and Related Topics pp 79– (1971)
[102] Gupta S. S., Biometrika 60 pp 403– (1973) · Zbl 0263.62031
[103] Gupta S. S., Multivariate Analysis-II pp 475– (1969)
[104] Gupta S. S., Annals of Mathematical Statistics 43 pp 814– (1972) · Zbl 0251.62016
[105] Gupta S. S., Reliability and Biometry pp 503– (1974)
[106] Gupta S. S., Reliability and Fault Tree Analysis pp 557– (1975)
[107] Gupta S. S., Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations (1979) · Zbl 0516.62030
[108] GuptaS. S. and SantnerT. J. (1973). On selection and ranking procedures - a restricted subset selection rule.Proceedings of the 39th Session of the International Statistical Institute, Vol. 45, Book I, 478–486.
[109] Gupta S. S., Communications in Statistics - Theory and Methods 9 pp 1277– (1980) · Zbl 0443.62015
[110] Gupta S. S., Annals of Mathematical Statistics 29 pp 235– (1958) · Zbl 0088.12601
[111] Gupta S. S., Contributions to Probability and Statistics pp 224– (1960)
[112] Gupta S. S., Essays in Probability and Statistics pp 327– (1970)
[113] Gupta S. S., Colloquia Mathematica Societatis JánQS Bolyai, 16: Topics in Information Theory pp 279– (1977)
[114] Gupta S. S., Operations Research Verfahren pp 49– (1977)
[115] Gupta S. S., Selecta Statistica Canadiana 6 pp 109– (1982)
[116] Gupta S. S., Statistical Decision Theory and Related Topics (1971)
[117] Gupta S. S., Statistics: Applications and New Directions pp 279– (1984)
[118] Hochberg Y., Communications in Statistics - Theory and Methods 10 pp 597– (1981) · Zbl 0481.62025
[119] Hocking R. R., Technometrics 9 pp 531– (1976)
[120] Hooper J. H., Annals of Statistics 7 pp 615– (1979) · Zbl 0408.62041
[121] Hsu J. C., On Some Decision-Theoretic Contributions to the Problem of Subset Selection (1977)
[122] Hsu J. C., Communications in Statistics - Theory and Methods 9 pp 1439– (1980) · Zbl 0454.62027
[123] Hsu J. C., Sankhyā 43 pp 235– (1981)
[124] Hsu J. C., Annals of Statistics 9 pp 1026– (1981) · Zbl 0474.62031
[125] Hsu J. C., Annals of Statistics 12 pp 1136– (1984) · Zbl 0557.62023
[126] Hsu T.-A., Communications in Statistics - Theory and Methods pp 411– (1982)
[127] Huang D.-Y., Communications in Statistics – Theory and Methods 5 pp 621– (1976) · Zbl 0376.62017
[128] Huang D.-Y., Journal of the Chinese Statistical Association 16 pp 5801– (1978)
[129] Huang D.-Y., Communications in Statistics - Theory and Methods 11 pp 751– (1982) · Zbl 0511.62077
[130] Huang D.-Y., Statistical Decision Theory and Related Topics - III 2 pp 1– (1982)
[131] Huang D.-Y., Journal of Statistical Planning and Inference 9 pp 63– (1984) · Zbl 0536.62011
[132] Huang D.-Y., Communications in Statistics - Theory and Methods 12 pp 341– (1983) · Zbl 0522.62019
[133] Jeyaratnam S., Design of Experiments: Ranking and Selection pp 287– (1984)
[134] Kiefer J., Multivariate Analysis-IV pp 143– (1977)
[135] Kiefer J., Journal of the American Statistical Association 72 pp 789– (1977)
[136] Kleijnen J. P. C., Statistical Techniques in Simulation 2 (1976)
[137] Krishnaiah P. R., Blanch Anniversary Volume pp 147– (1967)
[138] Krishnaiah P. R., Multivariate Analysis pp 477– (1966)
[139] Lehmann E. L., Mathematische Annalen 150 pp 268– (1963) · Zbl 0114.11203
[140] Lehmann E. L., Annals of Mathematical Statistics 34 pp 957– (1963) · Zbl 0203.21106
[141] Lorenzen T. J., Communications in Statistics - Theory and Methods 10 pp 101– (1981) · Zbl 0468.62024
[142] McCabe G. P., Journal of Statistical Computation and Simulation 3 pp 137– (1974) · Zbl 0294.62063
[143] McDonald G. C., Sankhyā 34 pp 53– (1972)
[144] McDonald G. C., Sankhyā 36 pp 261– (1974)
[145] Miescke K. J., Communications in Statistics - Theory and Methods 8 pp 1205– (1979) · Zbl 0407.62015
[146] Miescke K. J., Proceedings of the Seventh Conference on Probability Theory pp 259– (1984)
[147] Mishra S. N., Simultaneous selection of extreme populations: a subset selection approach (1983)
[148] Mosteller F., Annals of Mathematical Statistics 19 pp 58– (1948) · Zbl 0031.37102
[149] Nagel K., On Subset Selection Rules with Certain Optimality Properties (1970)
[150] Panchapakesan S., Statistical Decision Theory and Related Topics pp 275– (1971)
[151] Panchapakesan S., Bulletin of the Institute of Mathematical Statistics 2 pp 112– (1973)
[152] Panchapakesan S., Communications in Statistics - Theory and Methods 6 pp 1081– (1977)
[153] Patel J. K., Journal of the American Statistical Association 71 pp 143– (1976)
[154] Paulson E., Annals of Mathematical Statistics 23 pp 610– (1952) · Zbl 0047.38204
[155] du Preez J. P., South African Statistical Journal 19 pp 45– (1985)
[156] Puri M. L., Annals of Mathematical Statistics 40 pp 619– (1969) · Zbl 0174.21803
[157] Puri P. S., Sankhyā 30 pp 291– (1968)
[158] Putter J., Poultry Science 43 pp 1425– (1964)
[159] Putter J., Journal of Dairy Science 48 pp 747– (1965)
[160] Rizvi M. H., Technometrics 13 pp 355– (1971)
[161] Rizvi M. H., Annals of Mathematical Statistics 38 pp 1788– (1967) · Zbl 0214.46001
[162] Rizvi M. H., Annals of Mathematical Statistics 39 pp 2075– (1968) · Zbl 0177.46603
[163] Rizvi M. H., Annals of Mathematical Statistics 41 pp 1942– (1970) · Zbl 0245.62030
[164] Robbins H., Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability pp 131– (1951)
[165] Robbins H., Annals of Mathematical Statistics 41 pp 1397– (1970) · Zbl 0239.62025
[166] Santner T. J., A Restricted Subset Selection Approach to Ranking and Selection Problems (1973) · Zbl 0302.62011
[167] Santner T. J., Annals of Statistics 3 pp 334– (1975) · Zbl 0302.62011
[168] Seal K. C., Annals of Mathematical Statistics 26 pp 387– (1955) · Zbl 0065.12501
[169] Seal K. C., Calcutta Statistical Association Bulletin 7 pp 131– (1957)
[170] Seal K. C., Journal of the American Statistical Association 53 pp 164– (1958)
[171] Seal K. C., Calcutta Statistical Association Bulletin 8 pp 20– (1958)
[172] Somerville P. N., Journal of Statistical Computation and Simulation 19 pp 215– (1984)
[173] Studden W. J., Annals of Mathematical Statistics 38 pp 1072– (1967) · Zbl 0152.36102
[174] Swanepoel J. W., South African Statistical Journal 7 pp 11– (1973)
[175] Tamhane A. C., Sankhyā 38 pp 339– (1976)
[176] Tamhane A. C., Communications in Statistics - Theory and Methods 6 pp 1003– (1977)
[177] Tamhane A. C., Communications in Statistics - Theory and Methods 8 pp 337– (1979) · Zbl 0442.62028
[178] Thompson M. L., International Statistical Review 46 pp 1– (1978) · Zbl 0379.62055
[179] Thompson M. L., International Statistical Review 46 pp 129– (1978) · Zbl 0426.62046
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