Stein's two-stage procedure and exact consistency

Abstract

Stein (1945 and 1949) achieved the exact probability coverage for a fixed-width confidence interval estimation of a normal mean when the variance is unknown. To achieve this type of exact result of “consistency” (in the Chow—Robbins (1965) sense), we notice that the assumption of normal distributions for the population is not essential. It can be replaced by less restrictive conditions on independence and pivotal nature of some suitable statistics. This is our Theorem 1. Examples are provided from negative exponential, symmetric normal (Rao, 1973, pp. 196–198) and normal populations. A similar type of problem is discussed for inverse Gaussian (see Folks & Chhikara, 1978) parameters. Modified two-stage procedures are proposed along the lines of Mukhopadhyay (1980) and are shown to be asymptotically “first-order efficient” (in the Ghosh—Mukhopadhyay (1981) sense). We also develop and study some properties of two-stage fixed-width confidence intervals constructed along the lines of Birnbaum & Healy (1960).