A Stein-Type Two-Sample Procedure for Comparing Normal Means

In this article, we propose a Stein-type two-sample procedure for comparing the means of k(> 1) experimental normal populations among themselves and with the reference to the mean of a controlled normal population, when the variances of all (k + 1) populations are unequal and unknown. Our selection formulation follows closely to that by Bechoffer and Turnbull (1978), who considered the comparison of k normal means with a specific nonrandom standard value, when the variances are either known or unknown and equal. Instead of comparing the k experimental populations to a nonrandom standard value, our comparison is made with reference to a random controlled normal population. Moreover, we broaden their assumption of equal unknown variances to unequal unknown variances. The proposed procedure satisfies two probability requirements: (1) the probability of selecting the control is at least prespecified (Formula presented.) when the largest experimental mean is significantly smaller than the mean of the control and (2) the probability of selecting the largest experimental mean is at least prespecified (Formula presented.) when the largest experimental mean is significantly largest than the second largest experimental mean and the mean of the control.