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

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 - alpha) + Ad(2) + o(d(2)) as d --> 0 where (1 - alpha) 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 - alpha) 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 tau*. 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 Op(tau*). 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.