Subset selection for the best of two populations : tables of the expected subset size

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Subset selection for the best of two populations : tables of the

expected subset size

Citation for published version (APA):

Laan, van der, P., & Eeden, van, C. (1993). Subset selection for the best of two populations : tables of the expected subset size. (Memorandum COSOR; Vol. 9301). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1993

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum

CaSaR

93-01

Subset selection for the best of two populations:

Tables of the expected subset size

P. van der Laan C. van Eeden

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Eindhoven University of Technology

Department of Mathematics and Computing Science

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Subset selection for the best of two populations:

Tables of the expected subset size

Paul van der Laan

Eindhoven University of Technology and

Constance van Eeden University of Britisch Colombia

Universite du Quebec

a

Montreal

Summary

Assume two independent populations are given. The associated independent random

vari-ables have Normal distributions with unknown expectations

fh

and (J2, respectively, and

known common variance (72. The selection goal of Gupta's subset selection for two

pop-ulations is to select a non-empty subset which contains the best, in the sense of largest

expectation, population with confidence level p*(!

<

P*

<

1). In Van der Laan and Van

Eeden (1992) a generalized selection goal has been introduced and investigated. Inthis report

extended tables with values of the expected subset size are given.

AMS Subject classification: Primary 62F07; secondary 62E15.

Key Words and Phrases: subset selection, loss function, expected subset size, generalized selection goal.

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1. Introduction

Assume two populations 11"1 and 11"2 are given. The related independent random variables,

which may be sample means, are denoted byXl and X2 , respectively. The random variable

Xi (i

=

1,2) has a Normal distribution with unknown expectation ()i and known variance (72.

Without loss of generality we can assume (72 = 1. The ordered expectations are denoted by

8[1] ~ 8[2]' The best population is defined as the population with parameter()(2]. We assume there is not a tie. The subset selection procedure selects a subset, non-empty and as small as possible, with the probability requirement that the probability of a correct selection is at

least P*

(!

<

P*

<

1). A correct selection means that the best population is an element of

the subset. The selection rule of Gupta (1965) runs as follows. Select population 11"i in the

subset if and only ifXi

2:

~axXj - d (i= 1,2). The selection constantdmust be determined

3=1,2

such that P(CS)

2:

P* for all possible parameter configurations.

In Van der Laan and Van Eeden (1992) a new criterion for subset selection in terms of a

natural loss function has been considered. Indicating the subset size by S, in Van der Laan

and Van Eeden (1992) it has been proved that

ES

=

<I?

(~)

+

<I?

(C

+

P)

v'2

v'2'

where P :=

18

1 -

8

2

1,

for the selection rule

11"1 in subset iffXl - x2

>

C 11"2 in subset iffXl - X2

<

-C

11"1 and 11"2 in subset iff -c ~ Xl - X2 ~ C ,

with c

2:

o.

Ifc= 0, then ES= 1. Furtheron ES= 2 for c= 00.

Inorder to fulfill certain requirements, e.g. ES~ 1

+

6 for some p's, a limited table with ES

for some values of c and p has been inserted in Van der Laan and Van Eeden (1992).

In the extended table 1 the expected subset size ES is given in four decimal places and for

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Table 1

The expected subset size

ES=

~ (~) +~

(C+JL)

v'2

v'2'

where

JL

=

10

1 -

0

2

1,

for C

=

0.1(0.1)3.0 and

JL

= 0(0.1)3.0.

C

JL

ES C

_JL

ES C

_JL

ES 0.1 0.0 1.0564 0.2 0.0 1.1125 0.3 0.0 1.1680 0.1 0.1 1.0562 0.2 0.1 1.1122 0.3 0.1 1.1676 0.1 0.2 1.0558 0.2 0.2 1.1114 0.3 0.2 1.1663 0.1 0.3 1.0551 0.2 0.3 1.1100 0.3 0.3 1.1643 0.1 0.4 1.0542 0.2 0.4 1.1081 0.3 0.4 1.1615 0.1 0.5 1.0530 0.2 0.5 1.1057 0.3 0.5 1.1580 0.1 0.6 1.0515 0.2 0.6 1.1028 0.3 0.6 1.1537 0.1 0.7 1.0499 0.2 0.7 1.0996 0.3 0.7 1.1489 0.1 0.8 1.0480 0.2 0.8 1.0959 0.3 0.8 1.1435 0.1 0.9 1.0461 0.2 0.9 1.0920 0.3 0.9 1.1376 0.1 1.0 1.0439 0.2 1.0 1.0877 0.3 1.0 1.1313 0.1 1.1 1.0417 0.2 1.1 1.0833 0.3 1.1 1.1247 0.1 1.2 1.0394 0.2 1.2 1.0787 0.3 1.2 1.1178 0.1 1.3 1.0370 0.2 1.3 1.0739 0.3 1.3 1.1108 0.1 1.4 1.0346 0.2 1.4 1.0691 0.3 1.4 1.1037 0.1 1.5 1.0321 0.2 1.5 1.0643 0.3 1.5 1.0965 0.1 1.6 1.0298 0.2 1.6 1.0596 0.3 1.6 1.0894 0.1 1.7 1.0274 0.2 1.7 1.0549 0.3 1.7 1.0824 0.1 1.8 1.0251 0.2 1.8 1.0503 0.3 1.8 1.0756 0.1 1.9 1.0229 0.2 1.9 1.0459 0.3 1.9 1.0691 0.1 2.0 1.0208 0.2 2.0 1.0416 0.3 2.0 1.0627 0.1 2.1 1.0188 0.2 2.1 1.0376 0.3 2.1 1.0567 0.1 2.2 1.0168 0.2 2.2 1.0338 0.3 2.2 1.0510 0.1 2.3 1.0151 0.2 2.3 1.0302 0.3 2.3 1.0457 0.1 2.4 1.0134 0.2 2.4 1.0269 0.3 2.4 1.0407 0.1 2.5 1.0118 0.2 2.5 1.0238 0.3 2.5 1.0360 0.1 2.6 1.0104 0.2 2.6 1.0210 0.3 2.6 1.0318 0.1 2.7 1.0091 0.2 2.7 1.0184 0.3 2.7 1.0279 0.1 2.8 1.0080 0.2 2.8 1.0160 0.3 2.8 1.0244 0.1 2.9 1.0069 0.2 2.9 1.0139 0.3 2.9 1.0212 0.1 3.0 1.0060 0.2 3.0 1.0120 0.3 3.0 1.0183

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c _J-t ES c _J-t ES c _J-t ES 0.4 0.0 1.2227 0.5 0.0 1.2763 0.6 0.0 1.3286 0.4 0.1 1.2222 0.5 0.1 1.2757 0.6 0.1 1.3279 0.4 0.2 1.2205 0.5 0.2 1.2737 0.6 0.2 1.3255 0.4 0.3 1.2179 0.5 0.3 1.2704 0.6 0.3 1.3217 0.4 0.4 1.2142 0.5 0.4 1.2659 0.6 0.4 1.3165 0.4 0.5 1.2096 0.5 0.5 1.2602 0.6 0.5 1.3098 0.4 0.6 1.2040 0.5 0.6 1.2535 0.6 0.6 1.3019 0.4 0.7 1.1977 0.5 0.7 1.2457 0.6 0.7 1.2928 0.4 0.8 1.1906 0.5 0.8 1.2370 0.6 0.8 1.2827 0.4 0.9 1.1829 0.5 0.9 1.2275 0.6 0.9 1.2716 0.4 1.0 1.1746 0.5 1.0 1.2174 0.6 1.0 1.2597 0.4 1.1 1.1659 0.5 1.1 1.2067 0.6 1.1 1.2472 0.4 1.2 1.1569 0.5 1.2 1.1956 0.6 1.2 1.2341 0.4 1.3 1.1476 0.5 1.3 1.1843 0.6 1.3 1.2208 0.4 1.4 1.1382 0.5 1.4 1.1727 0.6 1.4 1.2072 0.4 1.5 1.1288 0.5 1.5 1.1611 0.6 1.5 1.1935 0.4 1.6 1.1194 0.5 1.6 1.1496 0.6 1.6 1.1799 0.4 1.7 1.1102 0.5 1.7 1.1382 0.6 1.7 1.1664 0.4 1.8 1.1012 0.5 1.8 1.1270 0.6 1.8 1.1532 0.4 1.9 1.0925 0.5 1.9 1.1163 0.6 1.9 1.1404 0.4 2.0 1.0841 0.5 2.0 1.1059 0.6 2.0 1.1281 0.4 2.1 1.0761 0.5 2.1 1.0960 0.6 2.1 1.1163 0.4 2.2 1.0685 0.5 2.2 1.0865 0.6 2.2 1.1051 0.4 2.3 1.0614 0.5 2.3 1.0777 0.6 2.3 1.0945 0.4 2.4 1.0548 0.5 2.4 1.0694 0.6 2.4 1.0846 0.4 2.5 1.0486 0.5 2.5 1.0617 0.6 2.5 1.0754 0.4 2.6 1.0429 0.5 2.6 1.0546 0.6 2.6 1.0668 0.4 2.7 1.0377 0.5 2.7 1.0481 0.6 2.7 1.0590 0.4 2.8 1.0330 0.5 2.8 1.0421 0.6 2.8 1.0518 0.4 2.9 1.0287 0.5 2.9 1.0367 0.6 2.9 1.0453 0.4 3.0 1.0249 0.5 3.0 1.0319 0.6 3.0 1.0394

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c J1, ES c J1, ES c J1, ES 0.7 0.0 1.3794 0.8 0.0 1.4284 0.9 0.0 1.4755 0.7 0.1 1.3785 0.8 0.1 1.4274 0.9 0.1 1.4744 0.7 0.2 1.3759 0.8 0.2 1.4246 0.9 0.2 1.4714 0.7 0.3 1.3716 0.8 0.3 1.4198 0.9 0.3 1.4662 0.7 0.4 1.3657 0.8 0.4 1.4133 0.9 0.4 1.4592 0.7 0.5 1.3582 0.8 0.5 1.4050 0.9 0.5 1.4503 0.7 0.6 1.3492 0.8 0.6 1.3951 0.9 0.6 1.4396 0.7 0.7 1.3389 0.8 0.7 1.3838 0.9 0.7 1.4273 0.7 0.8 1.3274 0.8 0.8 1.3710 0.9 0.8 1.4135 0.7 0.9 1.3148 0.8 0.9 1.3571 0.9 0.9 1.3985 0.7 1.0 1.3013 0.8 1.0 1.3422 0.9 1.0 1.3823 0.7 1.1 1.2871 0.8 1.1 1.3264 0.9 1.1 1.3651 0.7 1.2 1.2723 0.8 1.2 1.3100 0.9 1.2 1.3472 0.7 1.3 1.2570 0.8 1.3 1.2931 0.9 1.3 1.3288 0.7 1.4 1.2415 0.8 1.4 1.2758 0.9 1.4 1.3099 0.7 1.5 1.2259 0.8 1.5 1.2584 0.9 1.5 1.2908 0.7 1.6 1.2103 0.8 1.6 1.2410 0.9 1.6 1.2718 0.7 1.7 1.1949 0.8 1.7 1.2237 0.9 1.7 1.2528 0.7 1.8 1.1798 0.8 1.8 1.2068 0.9 1.8 1.2341 0.7 1.9 1.1651 0.8 1.9 1.1902 0.9 1.9 1.2159 0.7 2.0 1.1509 0.8 2.0 1.1742 0.9 2.0 1.1982 0.7 2.1 1.1372 0.8 2.1 1.1588 0.9 2.1 1.1811 0.7 2.2 1.1243 0.8 2.2 1.1442 0.9 2.2 1.1648 0.7 2.3 1.1120 0.8 2.3 1.1302 0.9 2.3 1.1493 0.7 2.4 1.1005 0.8 2.4 1.1171 0.9 2.4 1.1346 0.7 2.5 1.0897 0.8 2.5 1.1049 0.9 2.5 1.1208 0.7 2.6 1.0797 0.8 2.6 1.0934 0.9 2.6 1.1080 0.7 2.7 1.0705 0.8 2.7 1.0829 0.9 2.7 1.0961 0.7 2.8 1.0621 0.8 2.8 1.0732 0.9 2.8 1.0851 0.7 2.9 1.0544 0.8 2.9 1.0643 0.9 2.9 1.0750 0.7 3.0 1.0475 0.8 3.0 1.0563 0.9 3.0 1.0659

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c J.L ES c J.L ES c J.L ES 1.0 0.0 1.5205 1.1 0.0 1.5633 1.2 0.0 1.6039 1.0 0.1 1.5194 1.1 0.1 1.5622 1.2 0.1 1.6027 1.0 0.2 1.5161 1.1 0.2 1.5588 1.2 0.2 1.5991 1.0 0.3 1.5107 1.1 0.3 1.5531 1.2 0.3 1.5933 1.0 0.4 1.5032 1.1 0.4 1.5453 1.2 0.4 1.5852 1.0 0.5 1.4937 1.1 0.5 1.5354 1.2 0.5 1.5750 1.0 0.6 1.4824 1.1 0.6 1.5235 1.2 0.6 1.5628 1.0 0.7 1.4693 1.1 0.7 1.5098 1.2 0.7 1.5486 1.0 0.8 1.4547 1.1 0.8 1.4944 1.2 0.8 1.5327 1.0 0.9 1.4386 1.1 0.9 1.4776 1.2 0.9 1.5152 1.0 1.0 1.4213 1.1 1.0 1.4594 1.2 1.0 1.4963 1.0 1.1 1.4030 1.1 1.1 1.4401 1.2 1.1 1.4762 1.0 1.2 1.3839 1.1 1.2 1.4199 1.2 1.2 1.4552 1.0 1.3 1.3641 1.1 1.3 1.3989 1.2 1.3 1.4333 1.0 1.4 1.3438 1.1 1.4 1.3775 1.2 1.4 1.4108 1.0 1.5 1.3233 1.1 1.5 1.3557 1.2 1.5 1.3879 1.0 1.6 1.3027 1.1 1.6 1.3337 1.2 1.6 1.3648 1.0 1.7 1.2822 1.1 1.7 1.3118 1.2 1.7 1.3417 1.0 1.8 1.2619 1.1 1.8 1.2902 1.2 1.8 1.3187 1.0 1.9 1.2421 1.1 1.9 1.2689 1.2 1.9 1.2961 1.0 2.0 1.2228 1.1 2.0 1.2481 1.2 2.0 1.2740 1.0 2.1 1.2041 1.1 2.1 1.2279 1.2 2.1 1.2524 1.0 2.2 1.1862 1.1 2.2 1.2085 1.2 2.2 1.2316 1.0 2.3 1.1692 1.1 2.3 1.1900 1.2 2.3 1.2117 1.0 2.4 1.1530 1.1 2.4 1.1723 1.2 2.4 1.1926 1.0 2.5 1.1378 1.1 2.5 1.1556 1.2 2.5 1.1745 1.0 2.6 1.1235 1.1 2.6 1.1400 1.2 2.6 1.1575 1.0 2.7 1.1102 1.1 2.7 1.1253 1.2 2.7 1.1415 1.0 2.8 1.0979 1.1 2.8 1.1118 1.2 2.8 1.1266 1.0 2.9 1.0866 1.1 2.9 1.0992 1.2 2.9 1.1128 1.0 3.0 1.0763 1.1 3.0 1.0877 1.2 3.0 1.1001

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c _JL ES c _JL ES c JL ES 1.3 0.0 1.6420 1.4 0.0 1.6778 1.5 0.0 1.7112 1.3 0.1 1.6408 1.4 0.1 1.6766 1.5 0.1 1.7100 1.3 0.2 1.6372 1.4 0.2 1.6730 1.5 0.2 1.7063 1.3 0.3 1.6313 1.4 0.3 1.6670 1.5 0.3 1.7004 1.3 0.4 1.6231 1.4 0.4 1.6587 1.5 0.4 1.6921 1.3 0.5 1.6126 1.4 0.5 1.6482 1.5 0.5 1.6816 1.3 0.6 1.6001 1.4 0.6 1.6355 1.5 0.6 1.6690 1.3 0.7 1.5857 1.4 0.7 1.6209 1.5 0.7 1.6543 1.3 0.8 1.5694 1.4 0.8 1.6044 1.5 0.8 1.6378 1.3 0.9 1.5515 1.4 0.9 1.5862 1.5 0.9 1.6195 1.3 1.0 1.5321 1.4 1.0 1.5665 1.5 1.0 1.5996 1.3 1.1 1.5114 1.4 1.1 1.5454 1.5 1.1 1.5784 1.3 1.2 1.4896 1.4 1.2 1.5232 1.5 1.2 1.5559 1.3 1.3 1.4670 1.4 1.3 1.5001 1.5 1.3 1.5324 1.3 1.4 1.4437 1.4 1.4 1.4761 1.5 1.4 1.5080 1.3 1.5 1.4199 1.4 1.5 1.4517 1.5 1.5 1.4831 1.3 1.6 1.3958 1.4 1.6 1.4268 1.5 1.6 1.4576 1.3 1.7 1.3717 1.4 1.7 1.4018 1.5 1.7 1.4319 1.3 1.8 1.3476 1.4 1.8 1.3768 1.5 1.8 1.4062 1.3 1.9 1.3239 1.4 1.9 1.3520 1.5 1.9 1.3805 1.3 2.0 1.3005 1.4 2.0 1.3276 1.5 2.0 1.3552 1.3 2.1 1.2777 1.4 2.1 1.3036 1.5 2.1 1.3302 1.3 2.2 1.2556 1.4 2.2 1.2803 1.5 2.2 1.3059 1.3 2.3 1.2343 1.4 2.3 1.2578 1.5 2.3 1.2822 1.3 2.4 1.2139 1.4 2.4 1.2361 1.5 2.4 1.2593 1.3 2.5 1.1945 1.4 2.5 1.2154 1.5 2.5 1.2374 1.3 2.6 1.1761 1.4 2.6 1.1957 1.5 2.6 1.2165 1.3 2.7 1.1588 1.4 2.7 1.1771 1.5 2.7 1.1966 1.3 2.8 1.1426 1.4 2.8 1.1596 1.5 2.8 1.1778 1.3 2.9 1.1275 1.4 2.9 1.1432 1.5 2.9 1.1602 1.3 3.0 1.1135 1.4 3.0 1.1280 1.5 3.0 1.1437

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c f.L ES c f.L ES c f.L ES 1.6 0.0 1.7421 1.7 0.0 1.7707 1.8 0.0 1.7969 1.6 0.1 1.7409 1.7 0.1 1.7695 1.8 0.1 1.7958 1.6 0.2 1.7374 1.7 0.2 1.7660 1.8 0.2 1.7924 1.6 0.3 1.7315 1.7 0.3 1.7603 1.8 0.3 1.7868 1.6 0.4 1.7233 1.7 0.4 1.7522 1.8 0.4 1.7790 1.6 0.5 1.7129 1.7 0.5 1.7420 1.8 0.5 1.7691 1.6 0.6 1.7004 1.7 0.6 1.7297 1.8 0.6 1.7571 1.6 0.7 1.6858 1.7 0.7 1.7154 1.8 0.7 1.7431 1.6 0.8 1.6694 1.7 0.8 1.6992 1.8 0.8 1.7273 1.6 0.9 1.6511 1.7 0.9 1.6812 1.8 0.9 1.7096 1.6 1.0 1.6313 1.7 1.0 1.6616 1.8 1.0 1.6903 1.6 1.1 1.6100 1.7 1.1 1.6405 1.8 1.1 1.6695 1.6 1.2 1.5875 1.7 1.2 1.6180 1.8 1.2 1.6474 1.6 1.3 1.5638 1.7 1.3 1.5944 1.8 1.3 1.6240 1.6 1.4 1.5393 1.7 1.4 1.5698 1.8 1.4 1.5995 1.6 1.5 1.5140 1.7 1.5 1.5444 1.8 1.5 1.5742 1.6 1.6 1.4882 1.7 1.6 1.5184 1.8 1.6 1.5481 1.6 1.7 1.4620 1.7 1.7 1.4919 1.8 1.7 1.5215 1.6 1.8 1.4357 1.7 1.8 1.4651 1.8 1.8 1.4945 1.6 1.9 1.4093 1.7 1.9 1.4383 1.8 1.9 1.4674 1.6 2.0 1.3832 1.7 2.0 1.4116 1.8 2.0 1.4402 1.6 2.1 1.3574 1.7 2.1 1.3850 1.8 2.1 1.4131 1.6 2.2 1.3321 1.7 2.2 1.3589 1.8 2.2 1.3863 1.6 2.3 1.3074 1.7 2.3 1.3333 1.8 2.3 1.3600 1.6 2.4 1.2835 1.7 2.4 1.3084 1.8 2.4 1.3342 1.6 2.5 1.2604 1.7 2.5 1.2843 1.8 2.5 1.3091 1.6 2.6 1.2383 1.7 2.6 1.2611 1.8 2.6 1.2849 1.6 2.7 1.2172 1.7 2.7 1.2388 1.8 2.7 1.2615 1.6 2.8 1.1971 1.7 2.8 1.2176 1.8 2.8 1.2392 1.6 2.9 1.1783 1.7 2.9 1.1975 1.8 2.9 1.2179 1.6 3.0 1.1605 1.7 3.0 1.1785 1.8 3.0 1.1977

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c _Il ES c _Il ES c _Il ES 1.9 0.0 1.8209 2.0 0.0 1.8427 2.1 0.0 1.8624 1.9 0.1 1.8198 2.0 0.1 1.8417 2.1 0.1 1.8615 1.9 0.2 1.8166 2.0 0.2 1.8386 2.1 0.2 1.8585 1.9 0.3 1.8112 2.0 0.3 1.8334 2.1 0.3 1.8536 1.9 0.4 1.8036 2.0 0.4 1.8262 2.1 0.4 1.8468 1.9 0.5 1.7941 2.0 0.5 1.8170 2.1 0.5 1.8381 1.9 0.6 1.7825 2.0 0.6 1.8059 2.1 0.6 1.8275 1.9 0.7 1.7689 2.0 0.7 1.7929 2.1 0.7 1.8150 1.9 0.8 1.7535 2.0 0.8 1.7781 2.1 0.8 1.8009 1.9 0.9 1.7364 2.0 0.9 1.7615 2.1 0.9 1.7850 1.9 1.0 1.7176 2.0 1.0 1.7433 2.1 1.0 1.7675 1.9 1.1 1.6972 2.0 1.1 1.7236 2.1 1.1 1.7484 1.9 1.2 1.6755 2.0 1.2 1.7024 2.1 1.2 1.7279 1.9 1.3 1.6525 2.0 1.3 1.6799 2.1 1.3 1.7061 1.9 1.4 1.6284 2.0 1.4 1.6562 2.1 1.4 1.6830 1.9 1.5 1.6032 2.0 1.5 1.6315 2.1 1.5 1.6589 1.9 1.6 1.5773 2.0 1.6 1.6059 2.1 1.6 1.6337 1.9 1.7 1.5508 2.0 1.7 1.5796 2.1 1.7 1.6077 1.9 1.8 1.5237 2.0 1.8 1.5526 2.1 1.8 1.5811 1.9 1.9 1.4964 2.0 1.9 1.5253 2.1 1.9 1.5539 1.9 2.0 1.4689 2.0 2.0 1.4977 2.1 2.0 1.5263 1.9 2.1 1.4414 2.0 2.1 1.4699 2.1 2.1 1.4985 1.9 2.2 1.4141 2.0 2.2 1.4423 2.1 2.2 1.4706 1.9 2.3 1.3872 2.0 2.3 1.4148 2.1 2.3 1.4428 1.9 2.4 1.3607 2.0 2.4 1.3877 2.1 2.4 1.4153 1.9 2.5 1.3348 2.0 2.5 1.3611 2.1 2.5 1.3881 1.9 2.6 1.3096 2.0 2.6 1.3351 2.1 2.6 1.3614 1.9 2.7 1.2852 2.0 2.7 1.3099 2.1 2.7 1.3353 1.9 2.8 1.2618 2.0 2.8 1.2855 2.1 2.8 1.3100 1.9 2.9 1.2394 2.0 2.9 1.2620 2.1 2.9 1.2856 1.9 3.0 1.2181 2.0 3.0 1.2395 2.1 3.0 1.2621

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c Jl, ES c Jl, ES c Jl, ES 2.2 0.0 1.8802 2.3 0.0 1.8961 2.4 0.0 1.9103 2.2 0.1 1.8793 2.3 0.1 1.8953 2.4 0.1 1.9095 2.2 0.2 1.8765 2.3 0.2 1.8927 2.4 0.2 1.9071 2.2 0.3 1.8719 2.3 0.3 1.8884 2.4 0.3 1.9031 2.2 0.4 1.8655 2.3 0.4 1.8823 2.4 0.4 1.8975 2.2 0.5 1.8572 2.3 0.5 1.8746 2.4 0.5 1.8903 2.2 0.6 1.8472 2.3 0.6 1.8652 2.4 0.6 1.8815 2.2 0.7 1.8354 2.3 0.7 1.8541 2.4 0.7 1.8711 2.2 0.8 1.8220 2.3 0.8 1.8414 2.4 0.8 1.8592 2.2 0.9 1.8068 2.3 0.9 1.8271 2.4 0.9 1.8458 2.2 1.0 1.7901 2.3 1.0 1.8112 2.4 1.0 1.8308 2.2 1.1 1.7718 2.3 1.1 1.7938 2.4 1.1 1.8143 2.2 1.2 1.7521 2.3 1.2 1.7750 2.4 1.2 1.7965 2.2 1.3 1.7311 2.3 1.3 1.7548 2.4 1.3 1.7772 2.2 1.4 1.7087 2.3 1.4 1.7333 2.4 1.4 1.7566 2.2 1.5 1.6852 2.3 1.5 1.7106 2.4 1.5 1.7348 2.2 1.6 1.6607 2.3 1.6 1.6868 2.4 1.6 1.7119 2.2 1.7 1.6353 2.3 1.7 1.6620 2.4 1.7 1.6878 2.2 1.8 1.6090 2.3 1.8 1.6363 2.4 1.8 1.6628 2.2 1.9 1.5821 2.3 1.9 1.6099 2.4 1.9 1.6370 2.2 2.0 1.5547 2.3 2.0 1.5828 2.4 2.0 1.6104 2.2 2.1 1.5270 2.3 2.1 1.5553 2.4 2.1 1.5833 2.2 2.2 1.4991 2.3 2.2 1.5275 2.4 2.2 1.5557 2.2 2.3 1.4711 2.3 2.3 1.4994 2.4 2.3 1.5277 2.2 2.4 1.4432 2.3 2.4 1.4714 2.4 2.4 1.4997 2.2 2.5 1.4156 2.3 2.5 1.4434 2.4 2.5 1.4715 2.2 2.6 1.3883 2.3 2.6 1.4157 2.4 2.6 1.4436 2.2 2.7 1.3616 2.3 2.7 1.3884 2.4 2.7 1.4158 2.2 2.8 1.3355 2.3 2.8 1.3617 2.4 2.8 1.3885 2.2 2.9 1.3102 2.3 2.9 1.3356 2.4 2.9 1.3617 2.2 3.0 1.2857 2.3 3.0 1.3102 2.4 3.0 1.3356

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c J.L ES c J.L ES c J.L ES 2.5 0.0 1.9229 2.6 0.0 1.9340 2.7 0.0 1.9438 2.5 0.1 1.9222 2.6 0.1 1.9333 2.7 0.1 1.9431 2.5 0.2 1.9199 2.6 0.2 1.9313 2.7 0.2 1.9413 2.5 0.3 1.9162 2.6 0.3 1.9279 2.7 0.3 1.9382 2.5 0.4 1.9111 2.6 0.4 1.9232 2.7 0.4 1.9339 2.5 0.5 1.9044 2.6 0.5 1.9170 2.7 0.5 1.9283 2.5 0.6 1.8963 2.6 0.6 1.9095 2.7 0.6 1.9214 2.5 0.7 1.8866 2.6 0.7 1.9006 2.7 0.7 1.9132 2.5 0.8 1.8755 2.6 0.8 1.8903 2.7 0.8 1.9038 2.5 0.9 1.8629 2.6 0.9 1.8787 2.7 0.9 1.8930 2.5 1.0 1.8489 2.6 1.0 1.8656 2.7 1.0 1.8809 2.5 1.1 1.8334 2.6 1.1 1.8511 2.7 1.1 1.8674 2.5 1.2 1.8166 2.6 1.2 1.8353 2.7 1.2 1.8527 2.5 1.3 1.7983 2.6 1.3 1.8181 2.7 1.3 1.8366 2.5 1.4 1.7788 2.6 1.4 1.7996 2.7 1.4 1.8191 2.5 1.5 1.7579 2.6 1.5 1.7798 2.7 1.5 1.8004 2.5 1.6 1.7359 2.6 1.6 1.7588 2.7 1.6 1.7805 2.5 1.7 1.7127 2.6 1.7 1.7366 2.7 1.7 1.7593 2.5 1.8 1.6885 2.6 1.8 1.7133 2.7 1.8 1.7370 2.5 1.9 1.6634 2.6 1.9 1.6890 2.7 1.9 1.7136 2.5 2.0 1.6374 2.6 2.0 1.6637 2.7 2.0 1.6892 2.5 2.1 1.6108 2.6 2.1 1.6377 2.7 2.1 1.6640 2.5 2.2 1.5836 2.6 2.2 1.6110 2.7 2.2 1.6379 2.5 2.3 1.5559 2.6 2.3 1.5837 2.7 2.3 1.6111 2.5 2.4 1.5279 2.6 2.4 1.5560 2.7 2.4 1.5838 2.5 2.5 1.4998 2.6 2.5 1.5280 2.7 2.5 1.5561 2.5 2.6 1.4717 2.6 2.6 1.4999 2.7 2.6 1.5281 2.5 2.7 1.4436 2.6 2.7 1.4717 2.7 2.7 1.4999 2.5 2.8 1.4159 2.6 2.8 1.4437 2.7 2.8 1.4718 2.5 2.9 1.3886 2.6 2.9 1.4160 2.7 2.9 1.4437 2.5 3.0 1.3618 2.6 3.0 1.3886 2.7 3.0 1.4160

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c _J.L ES c _J.L ES c J.L ES 2.8 0.0 1.9523 2.9 0.0 1.9597 3.0 0.0 1.9661 2.8 0.1 1.9517 2.9 0.1 1.9592 3.0 0.1 1.9657 2.8 0.2 1.9501 2.9 0.2 1.9577 3.0 0.2 1.9643 2.8 0.3 1.9473 2.9 0.3 1.9552 3.0 0.3 1.9621 2.8 0.4 1.9433 2.9 0.4 1.9516 3.0 0.4 1.9589 2.8 0.5 1.9382 2.9 0.5 1.9471 3.0 0.5 1.9548 2.8 0.6 1.9320 2.9 0.6 1.9414 3.0 0.6 1.9497 2.8 0.7 1.9246 2.9 0.7 1.9346 3.0 0.7 1.9436 2.8 0.8 1.9159 2.9 0.8 1.9268 3.0 0.8 1.9365 2.8 0.9 1.9060 2.9 0.9 1.9177 3.0 0.9 1.9283 2.8 1.0 1.8948 2.9 1.0 1.9075 3.0 1.0 1.9190 2.8 1.1 1.8824 2.9 1.1 1.8961 3.0 1.1 1.9086 2.8 1.2 1.8687 2.9 1.2 1.8835 3.0 1.2 1.8970 2.8 1.3 1.8537 2.9 1.3 1.8696 3.0 1.3 1.8842 2.8 1.4 1.8374 2.9 1.4 1.8544 3.0 1.4 1.8701 2.8 1.5 1.8198 2.9 1.5 1.8380 3.0 1.5 1.8548 2.8 1.6 1.8010 2.9 1.6 1.8203 3.0 1.6 1.8383 2.8 1.7 1.7809 2.9 1.7 1.8014 3.0 1.7 1.8206 2.8 1.8 1.7597 2.9 1.8 1.7812 3.0 1.8 1.8016 2.8 1.9 1.7373 2.9 1.9 1.7599 3.0 1.9 1.7814 2.8 2.0 1.7139 2.9 2.0 1.7375 3.0 2.0 1.7600 2.8 2.1 1.6894 2.9 2.1 1.7140 3.0 2.1 1.7376 2.8 2.2 1.6641 2.9 2.2 1.6895 3.0 2.2 1.7141 2.8 2.3 1.6380 2.9 2.3 1.6642 3.0 2.3 1.6896 2.8 2.4 1.6112 2.9 2.4 1.6381 3.0 2.4 1.6642 2.8 2.5 1.5839 2.9 2.5 1.6113 3.0 2.5 1.6381 2.8 2.6 1.5562 2.9 2.6 1.5839 3.0 2.6 1.6113 2.8 2.7 1.5281 2.9 2.7 1.5562 3.0 2.7 1.5840 2.8 2.8 1.5000 2.9 2.8 1.5282 3.0 2.8 1.5562 2.8 2.9 1.4718 2.9 2.9 1.5000 3.0 2.9 1.5282 2.8 3.0 1.4437 2.9 3.0 1.4718 3.0 3.0 1.5000

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References

Gupta, 8.8. (1965). On some multiple decision (selection and ranking) rules. Technometrics 7, 225-245.

Van der Laan and Van Eeden, C. (1992). 8ubset selection with a generalized selection goal based on a loss function. Unpublished manuscript. To be submitted for publication.

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List of COSOR-memoranda - 1993 Number 93-01 Month January Author P. v.d. Laan C. v. Eeden Title

Subset selection for the best of two populations: Tables of the expected subset size