Optimal Bayes procedures for selecting the better of two Bernoulli populations
Let \(\pi_ 1\) and \(\pi_ 2\) be two Bernoulli populations with probabilities of success \(p_ 1\) and \(p_ 2\), respectively. The authors consider a class of sequential procedures C(n) for selecting the population with the larger success probability. The class C(n) is defined by a sampling rule R which requires that at most n observations be taken from any one of the two populations, and a stopping rule S which requires that the sampling be stopped at the first stage m at which either inequality \(z_{1,m}\geq z_{2,m}+n-n_{2,m}\) or \(z_{2m}\geq z_{1,m}+n-n_{1,m}\) holds, where \(n_{i,m}(z_{i,m})\) denotes the total number of observations (successes) from population \(\pi_ i\) \((i=1,2)\). At termination, the population with the longer number of successes is selected. Assuming that \(p_ 1\) and \(p_ 2\) have independent Beta prior densities an optimal procedure in C(n) is given which minimizes E(N), the expected total number of observations until the sampling terminates. Empirical procedures as approximations to the optimal Bayes procedure are given. It is conjectured that the empirical procedures which use identical symmetric Beta priors for \(p_ 1\) and \(p_ 2\) are minimax within the class C(n).
adaptive sequential selection procedures
scientific article; zbMATH DE number 4009539
scientific article