A predictive approach for the selection of a fixed number of good treatments

Abstract

This paper offers a predictive approach for the selection of a fixed number (= t) of treatments from k treatments with the goal of controlling for predictive losses. For the ith treatment, independent observations X(ij) (j = 1, 2, ..., n) can be observed where X(ij)'s are normally distributed N(theta(i);sigma2). The ranked values of theta(i)'s and X(i)BAR's are theta(1) less-than-or-equal-to ... less-than-or-equal-to theta((k)) and X[1]BAR less-than-or-equal-to ... less-than-or-equal-to X[k]BAR and the selected subset S = {[k], [k - 1], ... , [k - t + 1]) will be considered. This paper distinguishes between two types of loss functions. A type I loss function associated with a selected subset S is the loss in utility from the selector's view point and is a function of theta(i) with i is-an-element-of S. A type II loss function associated with S measures the unfairness in the selection from candidates' viewpoint and is a function of theta(i) with i is-an-element-of S. This paper shows that under mild assumptions on the loss functions S is optimal and provides the necessary formulae for choosing n so that the two types of loss can be controlled individually or simultaneously with a high probability. Predictive bounds for the losses are provided. Numerical examples support the usefulness of the predictive approach over the design of experiment approach.