This paper considers a \(k\)-stage sequential estimation of the mean vector \(\mu\) of a normal distribution. The \(k\)-stage sequential procedure is a generalization of W. Liu [J. Stat. Plann. Inference 65, No. 1, 109–127 (1997; Zbl 0915.62068)] for the multivariate normal population \(N(\mu, \sigma^2\Sigma)\), where the variance \(\sigma^2\) is unknown and the matrix \(\Sigma\) is known. The authors evaluate the regret in minimum risk point estimation and prove that the regret is negative. The authors also obtain the second-order approximation of confidence region estimation. These theoretical results are verified through synthetic and Fisher’s Iris data sets.