Multivariate sequential point estimation

For a multivariate normal distribution with unknown mean vector and unknown dispersion matrix, a sequential procedure for estimating the unknown mean vector is suggested. The procedure is shown to be asymptotically "risk efficient" in the sense of Starr (Ann. Math. Statist. (1966), 1173-1185), and the asymptotic order of the "regret" (see Starr and Woodroofe, Proc. Nat. Acad. Sci. 63 (1969), 285-288) is given. Moderate sample behaviour of the procedure using Monte-Carlo techniques is also studied. Finally, the asymptotic normality of the stopping time is proved.