On sequential fixed-width confidence intervals for the mean and second-order expansions of the associated coverage probabilities

Abstract

In order to construct fixed-width (2d) confidence intervals for the mean of an unknown distribution function F, a new purely sequential sampling strategy is proposed first. The approach is quite different from the more traditional methodology of Chow and Robbins (1965, Ann. Math. Statist., 36, 457–462). However, for this new procedure, the coverage probability is shown (Theorem 2.1) to be at least (1-α)+Ad ²+o(d²) as d→0 where (1-α) is the preassigned level of confidence and A is an appropriate functional of F, under some regularity conditions on F. The rates of convergence of the coverage probability to (1-α) obtained by Csenki (1980, Scand. Actuar. J., 107–111) and Mukhopadhyay (1981, Comm. Statist. Theory Methods, 10, 2231–2244) were merely O(d^1/2-q), with 0<q<1/2, under the Chow-Robbins stopping time τ^*. It is to be noted that such considerable sharpening of the rate of convergence of the coverage probability is achieved even though the new stopping variable is O_p(τ^*). An accelerated version of the stopping rule is also provided together with the analogous second-order characteristics. In the end, an example is given for the mean estimation problem of an exponential distribution.