A new formulation for the multinomial selection problem

The two usual approaches to a multinomial ranking and selection problem (for selecting the t best cells out of k) are combined to form a new approach. In this new approach there is a preference zone (PZ) and an indifference zone (IZ), and the concept of a correct selection (CS) is defined differently in each of these zones. Lower bounds for the probability of correct selection P(CS) are then guaranteed in each of these zones using a single experiment. The procedure used is a composite of two inverse-sampling stopping-rules and depends on the ordered frequencies in the cells. The principle tool used to derive expressions for the P(CS), for the expected sample size EN, for the expected subset size ES and for other probabilities, is the Dirichlet integral (Type 2) which was recently tabulated. These Dirichlet integrals are also used to prove that the multiplicative slippage configuration is least favorable in the PZ and, for t = 1, that the equal parameter configuration is the worst configuration in the IZ. Numerical calculations are carried out for an illustrative example but extensive tables are not yet available.