True Population Mean Research Articles

Some extensions of Somerville's procedure for ranking means of normal populations

SUMMARY Somerville (1954) proposed a one-stage and a two-stage procedure for the selection of the population with the largest mean from a set of normal populations with unknown means and a common, known variance. The two-stage procedure eliminates one population after the first stage. For these procedures he assumed that a certain loss was incurred when an incorrect selection was made and also that a cost was due to sampling. The sample sizes which would minimize the maximum expected loss, the maximum being taken over all possible configurations of the true population means, were derived. For the special case of three populations he showed that the two-stage procedure, using appropriate allocations of observations between stages, has a smaller maximum expected loss than does the onestage procedure. This paper generalizes Somerville's formulation of the two-stage procedure to an arbitrary finite number of populations and presents results for the special case of four populations, where two two-stage procedures are considered. For the problem of selecting the population with the largest mean from a set of normal populations with unknown means and a common, known variance, Somerville (1954) proposed a one-stage and a two-stage procedure, which eliminates one population after the first stage. He assumed that a certain loss was incurred when an incorrect selection was made and also assumed a cost due to sampling. For these procedures, he derived the sample sizes which minimize the maximum expected loss, the maximum being taken over all possible configurations of the true population means. He showed, for the special case of three populations that the two-stage procedure, using appropriate allocations of observations between stages, has a smaller maximum expected loss than does the one-stage procedure. Other approaches to this problem have been made by Bechhofer, Sobel and Gupta (see Bechhofer, 1954, and Gupta, 1965). In this paper the formulation of Somerville's two-stage procedure is extended to an arbitrary, finite number of populations. For the case of four populations, the expected losses are analysed in detail and numerical results obtained. Two possible two-stage procedures are considered: in the first procedure, one population is discarded and in the second, two populations are discarded after the first stage. It is found that while both procedures, using appropriate allocations, have smaller maximum expected losses than does the corresponding one-stage procedure, which one of the two-stage procedures is the better depends on the allocation. These numerical results required the evaluation of certain multivariate normal integrals with arbitrary correlation matrices and arbitrary limits of integration. This was accomplished for 4- and 5-variate integrals using a method suggested