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Within the Bayesian framework, a robust two-stage procedure is proposed to deal with the problem of multivariate sequential estimation of the unknown mean vector with weighted squared error loss and fixed cost per observation. The proposed procedure depends on the present data but not on the distributions of outcome variables or the prior. It is shown that the proposed procedure shares the asymptotic properties with the optimal fixed-sample-size procedures for the arbitrary distributions and the asymptotically pointwise optimal procedures for the distributions of a multivariate exponential family with a large class of prior distributions. Simulation results indicate that the proposed two-stage procedure is robust to misspecification of the true parameters of the prior distribution and outperforms the purely sequential procedure and the asymptotically pointwise optimal procedure in terms of robustness.

Asymptotic optimality of a robust two-stage procedure in multivariate Bayes sequential estimation