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 (d 2 ) 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.