(Received December 3, 1997; revised June 18, 1998)
Abstract. Under a fairly general setup, we first modify the Stein-type two-stage methodology in order to incorporate some partial information in the form of a known and positive lower bound for the otherwise unknown nuisance parameter, theta (>0). This revised methodology is then shown to enjoy various customary second-order properties and expansions for functions of the associated stopping variable, under appropriate conditions. Such general machineries are later applied in different types of estimation as well as selection and ranking problems, giving a sense of a very broad spectrum of possibilities. This constitutes natural extensions of these authors' earlier paper (Mukhopadhyay and Duggan (1997a, Sankhya Ser. A, 59, 435-448)) on the fixed-width confidence interval estimation problem exclusively for the mean of a normal distribution having an unknown variance.
Key words and phrases: Consistency, second-order expansions, confidence regions, point estimation, regret, selection and ranking.