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Higher than second-order approximations via two-stage sampling. (English) Zbl 1081.62547
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 it is assumed that \(\sigma>\sigma_L\), where \(\sigma_L\) \((>0)\) is known. Under these circumstances, the seminal two-stage procedure of C. Stein [Ann. Math. Stat. 16, 243–258 (1945; Zbl 0060.30403); Econometrica 17, 77–78 (1949)] has been recently modified by N. Mukhopadhyay and W. T. Duggan [Sankhyā, Ser. A 59, No. 3, 435–448 (1997)], and that modified methodology was shown to enjoy asymptotic second-order characteristics, similar to those found by M. Woodroofe [Ann. Stat. 5, No. 5, 984–995 (1977; Zbl 0374.62081)] and M. Ghosh and N. Mukhopadhyay [Sankkhyā, Ser. A 43, No. 2, 220–227 (1981; Zbl 0509.62069)] in the case of 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 \(d\to 0\). 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 [loc. cit.] 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 to as the 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.

MSC:
62L12 Sequential estimation
62L05 Sequential statistical design
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