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

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 σ 2 1 ,σ 2 2 ,…,σ 2 k , 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. n 1 , n 2 ,…, n k such that Σ n i = N , required to guarantee a specified {δ ∗ , P ∗ } probability requirement. We prove that for k ≥3, the allocation n i ∝σ 2 i (which is convenient to implement in practice) is locally (and for k =3, numerically checked to be globally) optimal iff P ∗ ≤ P L or P ∗ ≥ P U , where P L and P U depend on the largest and the smallest relative variances, respect ively. For P L P ∗ P U , the globally optimal allocation is found by numerical search for k =3 and found to be approximately given by n i ∝σ i , the allocation that is known to be globally optimal for k =2.