# zbMATH — the first resource for mathematics

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