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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.

62L12 Sequential estimation
62L05 Sequential statistical design