OPTIMUM DESIGNS FOR SELECTING ONE OF TWO TREATMENTS, FIXED SAMPLE SIZE PLAN AND SEQUENTIAL PLAN IN A POPULATION WITH ONE PARAMETER EXPONENTIAL DISTRIBUTION

The problem to determine the optimum statistical procedure, in some specified sense, for choosing between two populations in the light of samples drawn from them, is very important in practical situation. Let us now enunciate a formulation of this problem in a more concrete form. In the first place, we shall treat the situation in which there are two treatments (denoted by A and B) to be performed on each of a large number of individuals (say N), and we shall assume that the effect of each treatment on each individual can be expressed in terms of one real number, and that it is distributed in accordance with a certain population distribution. Let xA be a treatment effect of A performed on an individual randomly drawn from the population associated with the treatment A, and let us assume that A-, is distributed with a distribution function FA(x; GA) under our circumstance, where OA is a parameter in the population distribution. While let xB be a treatment effect of B performed on an individual randomly drawn from the other population associated with the treatment B, and let us assume that x, is distributed with a distribution function FB(x; GB) under our circumstance, where 0, is a parameter in the population distribution. Under this general circumstance we shall be concerned with two types of the problem. In the problem of Type I, we set up the following assumptions concerning FA(x ; GA) and FB(x;t9B): (a) The types of the two distribution functions are the same and known to us. (b) The true values of OA and BB are both unknown to us. (c) Let us denote by d(OA, GB) an assigned function of 0A and 0, expressing a discrepancy between OA and GB. There exists an a priori distribution for the parameter o = d(0 A, GB) and it is a continuous type and known to us. On the other hand, in the problem of Type II, we set up the following assumptions concerning F A(x ; GA) and FB(x ; GB):