TY - JOUR

T1 - Designing experiments for selecting the largest normal mean when the variances are known and unequal

T2 - Optimal sample size allocation

AU - Bechhofer, Robert E.

AU - Hayter, Anthony J.

AU - Tamhane, Ajit C.

N1 - Funding Information:
* Research of this author was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University. ** This paper was completed while this author was at the University of Bath during the summer of 1988 on a Visiting Research Fellowship Grant from the British Science and Engineering Research Council.

PY - 1991/7

Y1 - 1991/7

N2 - We consider the problem of 'optimally' allocating a given total number, N, of observations to k≥2 normal populations having unknown means but known variances σ21,σ22,...,σ2k, when it is desired to select the population having the largest mean using a natural single-stage selection procedure based on sample means. Here 'optimal' allocation is one that maximizes the infimum of the probability of a correct selection (P(CS)) over the so-called preference zone of the parameter space (Bechhofer (1954)). The solution of this problem enables us to find the smallest possible N and the associated optimal allocation of the sample sizes, viz. n1,n2,...,nk such that Σ ni=N, required to guarantee a specified {δ*,P*} probability requirement. We prove that for k≥3, the allocation ni∝σ2i (which is convenient to implement in practice) is locally (and for k=3, numerically checked to be globally) optimal iff P*≤PL or P*≥PU, where PL and PU depend on the largest and the smallest relative variances, respect ively. For PL<P*<PU, the globally optimal allocation is found by numerical search for k=3 and found to be approximately given by ni∝σi, the allocation that is known to be globally optimal for k=2.

AB - We consider the problem of 'optimally' allocating a given total number, N, of observations to k≥2 normal populations having unknown means but known variances σ21,σ22,...,σ2k, when it is desired to select the population having the largest mean using a natural single-stage selection procedure based on sample means. Here 'optimal' allocation is one that maximizes the infimum of the probability of a correct selection (P(CS)) over the so-called preference zone of the parameter space (Bechhofer (1954)). The solution of this problem enables us to find the smallest possible N and the associated optimal allocation of the sample sizes, viz. n1,n2,...,nk such that Σ ni=N, required to guarantee a specified {δ*,P*} probability requirement. We prove that for k≥3, the allocation ni∝σ2i (which is convenient to implement in practice) is locally (and for k=3, numerically checked to be globally) optimal iff P*≤PL or P*≥PU, where PL and PU depend on the largest and the smallest relative variances, respect ively. For PL<P*<PU, the globally optimal allocation is found by numerical search for k=3 and found to be approximately given by ni∝σi, the allocation that is known to be globally optimal for k=2.

KW - Ranking and selection

KW - indifference-zone approach

KW - normal populations

KW - optimal allocation

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U2 - 10.1016/0378-3758(91)90067-O

DO - 10.1016/0378-3758(91)90067-O

M3 - Article

AN - SCOPUS:0011498769

VL - 28

SP - 271

EP - 289

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 3

ER -