In sequential estimation of the mean of a normal distribution, Mukhopadhyay and Duggan (1997) showed second-order properties of the Stein-type two-stage procedure under the assumption that the unknown variance has a known and positive lower bound. Mukhopadhyay and Duggan (1999) extended the above result to a fairly general setup. We consider the general two-stage procedure of Mukhopadhyay and Duggan (1999) and show asymptotic higher-order expansions of the average sample size and so on. It will be seen that our higher-order approximations are more accurate than the second-order approximations of Mukhopadhyay and Duggan (1999). As an example, our results are applied to the bounded risk estimation of the normal mean.
 Mukhopadhyay, N. and Duggan, W.T. (1997). Can a two-stage procedure enjoy second-order properties? Sankhya, Vol. 59, Ser. A, pp. 435-448.
 Mukhopadhyay, N. and Duggan, W. (1999). On a two-stage procedure having second-order properties with applications. Ann. Inst. Statist. Math., Vol. 51, pp. 621-636.
Keywords: sequential estimation; two-stage procedure; asymptotic efficiency; average sample size
Biography: Chikara Uno obtained his doctor's degree in 1995 at Niigata University of Japan. He has had a post at Akita University of Japan from 1996. Since his earliest paper was published in 1993, his work has been in the area of sequential analysis.