Designing experiments for selecting the largest normal mean when the variances are known and unequal: Optimal sample size allocation

Robert E. Bechhofer*, Anthony J. Hayter, Ajit C. Tamhane

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

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 σ2122,...,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.

Original languageEnglish (US)
Pages (from-to)271-289
Number of pages19
JournalJournal of Statistical Planning and Inference
Volume28
Issue number3
DOIs
StatePublished - Jul 1991

Keywords

  • Ranking and selection
  • indifference-zone approach
  • normal populations
  • optimal allocation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

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