A Method for Selecting the Size of the Initial Sample in Stein's Two Sample Procedure

Abstract

The use of an upper percentage point of the distribution of total sample size in conjunction with the expectation of the latter is proposed as a guide to the selection of the size of the initial sample when using some version of Stein's [5] two-sample procedure. It is a rapidly calculable function of the underlying population variance based on existing tables of the $\chi^2$ distribution. A rule-of-thumb is proposed to be used in making the actual selection of initial sample size. It is a simple matter to investigate the nature of the percentage point for different values of the variance over a limited range; a recommended conservative choice when the variance is not known is the selection of a large initial sample. Dantzig [2] proved the nonexistence of nontrivial tests of Student's hypothesis whose power was independent of the variance, a result extended by Stein to the general linear hypothesis. In the same paper Stein proposes a two-sample procedure whose power was independent of variance. The same two-sample method could be used to obtain a confidence interval for the mean of a normal distribution with predetermined length and confidence coefficient. Stein gave no specifications for the choice of the initial sample size, but Seelbinder [4] suggested that it be selected to minimize the expectation of the total sample. In a recent paper, Bechhofer, Dunnett and Sobel [1] used Stein's procedure for another application, noting that the variance of the total sample size increased as the size of the first sample decreased. An efficient choice of the size of the initial sample will hold the expectation of the sample small, and will further reduce the probability of an extremely large total sample. This note will explore the matter in further detail and show that an upper percentage point of the distribution of total sample size, when used inconjunction with the expectation, is a rapidly calculable guide to an efficient choice of the size of the first sample.