Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 2, 254-269

HIGHER THAN SECOND-ORDER APPROXIMATIONS VIA TWO-STAGE SAMPLING

By

NITIS MUKHOPADHYAY, * University of Connecticut, Storrs*

SUMMARY. We consider the classical fixed-width $(=2d)$ confidence interval estimation problem for the mean $\mu $ of a normal population whose variance $\sigma ^{2}$ is unknown, but {\em it is assumed that} $\sigma >\sigma _{L}$ {\em % where} $\sigma _{L}(>0)$ {\em is known.} Under these circumstances, the seminal two-stage procedure of Stein (1945, 1949) has been recently modified by Mukhopadhyay and Duggan (1997), and that modified methodology was shown to enjoy asymptotic second-order characteristics, similar to those found in Woodroofe (1977) and Ghosh and Mukhopadhyay (1981) in the case of the purely sequential estimation strategies, that is, expanding $E(N)$ and the coverage probability respectively up to the orders $o(1)$ and $o(d^{2})$ as $% d\rightarrow 0.$ In Theorem 1.1, we first obtain expansions of both lower and upper bounds of $E(N)$ up to the order $O(d^{6})$. In Theorem 1.2, we then provide expansions of the lower and upper bounds for the coverage probability associated with the two-stage procedure of Mukhopadhyay and Duggan (1997) up to the order $o(d^{4}),$ whereas Theorem 1.3 further sharpens this order of approximation to $O(d^{6})$. These results amount to what may be referred as the {\em third-order approximations and beyond} via double sampling. Such results are not available in the case of any existing purely sequential and other multistage estimation strategies.

*AMS (1991) subject classification.* Primary 62L12; secondary 62L05.

*Key words and phrases. *Fixed-width interval, consistency,
third-order expansions, higher-order expansions.