Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 3, 435-448

CAN A TWO STAGE PROCEDURE ENJOY SECOND-ORDER PROPERTIES?

By

NITIS MUKHOPADHYAY,

And

WILLIAM T. DUGGAN, University of Connecticut, Storrs

SUMMARY. We first consider the classical fixed-width confidence interval estimation problem for the mean m of a normal population whose variance s 2 is unknown, but a particular application scenario guides the experimenter to assume that s > s L where s L (>0) is known. The seminal two-stage methodology of Stein (1945, 1949), originally proposed when s (>0)is completely unknown, obviously needs major revisions since we wish to incorporate such added partial information regarding s in the determination of the final sample size. In the case of completely unknown s , Stein's (1945-1949) two-stage procedure is known to enjoy the consistency property, but it is not even first-order efficient. In the case when s > s L (>0), the revised two-stage procedure is shown to enjoy all the usual second order properties together with the consistency property. As a follow-up, we include a simulation exercise in the interval estimation scenario. The minimum risk point estimation problem for m is also discussed briefly in the same light.

AMS (1991) subject classification. 62L12, 62L05

Key words and phrases. Fixed -width intervals, consistency, second-order expansions, minimum risk, point estimation, regret expansion

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