On sequential fixed-size confidence regions for the mean vector

In order to construct a fixed-size confidence region for the mean vector of an unknown distribution function F, a new purely sequential sampling strategy is proposed first. For this new procedure, under some regularity conditions on F, the coverage probability is shown (Theorem 2.1) to be at least (1 - alpha) - B alpha(2)d(2) + o(d(2)) as d --> 0, where (1 - alpha) is the preassigned level of confidence; B is an appropriate functional of F, and 2d is the preassigned diameter of the proposed spherical confidence region for the mean Vector of F. An accelerated version of the slopping rule is also provided with the analogous second-order characteristics (Theorem 3.1). In the special case of a p-dimensional normal random variable, analogous purely sequential and accelerated sequential procedures as well as a three-stage procedure are briefly introduced together with their asymptotic second-order characteristics. (C) 1997 Academic Press.