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 functionF, a new purely sequential sampling strategy is proposed first. For this new procedure, under some regularity conditions onF, the coverage probability is shown (Theorem 2.1) to be at least (1−α)−Bα2d2+o(d2) asd→0, where (1−α) is the preassigned level of confidence,Bis an appropriate functional ofF, and 2dis the preassigned diameter of the proposed spherical confidence region for the mean vector ofF. An accelerated version of the stopping rule is also provided with the analogous second-order characteristics (Theorem 3.1). In the special case of ap-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.