Selecting the best population, provided it is better than a control: The unequal variance case

BECHHOFER and TURNBULL. (1978) proposed two procedures to compare k normal means with a standard and the procedures guarantee that (I) with probability at least P*₀ (specified), no category is selected when the best experimental category is sufficiently worst than the standard, and (2) with probability at least P*_l (specified), the best experimental category is selected when it is sufficiently better than the second best and the standard. For the case of common known variance, they studied a single-stage procedure. For the case of common unknown variance, they studied a two-stage procedure. Under the same formulation of BECHHOFER and TURNBULL (1978) and for the same selection goals (1) and (2) described above, WILCOX (1984a) proposed a procedure to the case of unknown and unequal variances, and supplied a table of the necessary constants to implement the procedure. This paper considers the case of unknown and unequal variances for the same formulation of Bechhofer and Turnbull, and Wilcox, but assumes that μ₀ is an unknown control. A two-stage procedure is proposed to solve the problem. A lower bound of the probability of a correct selection is derived and it takes the same form as the double integral appeared in RINOTT (1978) which was used for the ower bound of the probability of a correct selection for a different selection goal.