A CLOSED SEQUENTIAL PROCEDURE SELECTING THE BEST POPULATION IN A FAMILY OF POPULATIONS WITH ONE PARAMETER EXPONENTIAL DISTRIBUTIONS

The problem to select the best population in some specified sense from several assigned populations in the light of samples drawn from them is very important in practical situations. An experimenter who is faced with this problem may select the best population according to a certain statistical procedure in view of the informations supplied by samples drawn from these populations. In this connection several different statistical procedures have been introduced by many authors such as Bahadur [2], Bahadur and Robbins [3], Bechhofer [4], [5], Bechhofer and Blumenthal [6], Bechhofer, Dunnett and Sobel [7], Dunnett [8], Fabian [9], Girshick [10], Gupta and Sobel [11], Mosteller [13] and Paulson [14]～[18] Taylor and David [25], Truax [26]. They have discussed this problem from several different viewpoints such as slippage aspect grouping aspect and ranking aspect. Some of them appealed to sequential multiple decision procedures which were not necessary closed. On the other hand authors such as Armitage [1], Schneiderman [21], Schneiderman and Armitage [22], Sobel and Wald [23], Sobel [24] appealed to restricted or closed sequential procedures which, however, were not necessarily multiple decision procedure. It was Paulson [19], [20] who presented a class of closed sequential multiple decision procedures for a set of significance level a in $ 0 < alpha < 1 $ and for a certain configuration of population means for which the probability by which the best population (having the largest means) among several normal populations is selected is larger than the prescribed value $ 1-alpha $. The object of this paper is to generalize the results of Paulson [19], [20] in two directions. In the first place we shall be concerned with a more general class of population distributions, that is, one parameter exponential distributions. In the second place we shall discuss with a configuration of population parameters which are more general than that which Paulson [19], [20] did substantially consider. Such a generalized configuration of population parameters is indeed both subtle and necessary in our generalized set-up dealing with one parameter exponential distribution. It is shown in this paper that the essential aspect of our closed sequential statistical procedures (CSSP) can be established in the frame of additive family of sufficient statistics whose notion was introduced by Kitagawa [12]. It is also noted that the restrictions on population distributions are required to establish our results. However these restrictions are so mild that we can easily obtain various examples regarding normal distributions, $ x^2 $-distributions, Poisson distributions and binomial distributions