Bayes solutions of some simple statistical decision problems

Abstract

Bayes solutions are obtained for some specific simple statistical decision problems which may be stated in the following practical terms. The result of a trial of a weapon system is valued as success (1) or failure (0) and p , the true probablility of success is unknown. Prior to testing, it is assumed that any value of p in the interval (0,1) is equally likely. Take as an estimate of p , the fraction, Pe of a finite number of trials which result in success. At the conclusion of testing, the weapon system is accepted (terminal decision d1) if Pe.> 0 , and it is rejected (terminal decision d2) if Pe < 0 , where 0 is a number in the interval (o,1) chosen by the experimenter. If Pe= 0 , either terminal decision may be made. A wrong decision may be made in two ways: (i) if the weapon system is accepted and the true p lies in an interval (0, [infinity]), where [inifnity] > 0 or (ii) if the weapon system is rejected and the true p lies in an interval (1 - b, 1) where 0 < 1-b . In this paper we consider only the case where [infinity and [beta] have the same value which we denote as 0 (symmetric case). The cost of wrong decision is defined in terms of a symmetric weight function, W(p,d1; 0), i = 1,2; W is simple or linear as defined below.