An integrated approach to some ranking and selection problems

We refer to the two classical approaches to ranking and selection problems as the indifference zone approach and the subset selection approach. In this thesis, we integrate these two approaches in selecting (1) the population with the largest mean (the best population) among k normal populations with unknown variances; (2) the population associated with the largest population proportion (the best population) among k binomial populations assuming a common large sample size. In this integrated approach, the parameter space is divided into two disjoint subsets, namely the preference zone (PZ) and the indifference zone (IZ). The concept of correct selection is defined differently in each of these zones. In the PZ, we are required to select the best population for a correct selection ($CS\sb1$). In the IZ, we define any selected subset to be correct ($CS\sb2$) if it contains the best population. A Stein-type (Stein (1945)) two-stage selection procedure is proposed for the normal case with common and unknown variance. For the normal case with uncommon and unknown variances, a Dudewicz-Dalal-type two-stage selection procedure is proposed. A selection rule is also proposed for the large sample binomial case. Lower bounds and formulas for the probability of correct selection under PZ and the probability of correct selection under IZ are developed. It is shown that the least favorable configuration (LFC) in PZ is the slippage configuration, and the worst configuration (WC) in IZ is the equal parameter configuration for the unknown and equal variance normal case. For the unknown and unequal variances normal case, it is proven that the slippage configuration is the least favorable configuration (LFC) in PZ and simulation study is conducted to investigate the worst configuration in IZ. For the binomial case, it is shown that the equal parameter configuration is the worst configuration in IZ for the case of k = 2 and the least favorable configuration (LFC) in PZ is the slippage configuration. A set of sufficient condition is also given for the monotonicity of the probability of correct selection in the indifference zone for the binomial case. It is proven that all proposed procedures guarantee that the following probability requirements are met: (1) the probability of selecting the best population in the PZ is at least $P\sbsp{1}{*}$, and (2) the probability of selecting a subset which would contain the best population is at least $P\sbsp{2}{*}$ when the true parameters are in the IZ. The expected subset sizes for all procedures are studied. Bounds for the expected sample sizes are developed for the normal cases. Tables necessary to implement these procedures are provided. Numerical examples are given. Simulation studies are also conducted