Multiple Comparisons with the Best Treatment

Let π1, π2, …, π k be k ≥ 2 sources of observations (treatments, populations) and suppose the “goodness” of treatment π i is characterized by the size of an unknown real-valued parameter θ i . Let θ[k] = max1≤i≤k θ i . If π i is preferred to π j when θ i > θ j , the parameters δ i = θ[k] — θ i , i = 1, 2, …, k reflect in an inverse sense the “goodness” of each treatment relative to the “best” treatment. A general technique for obtaining simultaneous confidence intervals on the δ i is demonstrated with several examples. This technique can be applied in any setting where comparison-with-control intervals can be computed regarding any π j as the control. These results have special importance in ranking and selection problems in that the process of generating upper bounds on the δ i generates traditional confidence statements of both the indifference zone and the subset selection schools, simultaneously, as established by Hsu (1981).