On unbiased and improved loss estimation for the mean of a multivariate normal distribution with unknown variance

There is now a sizeable literature dealing with point estimation using Stein-type estimators. As discussed in Rukhin (In: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics, Vol. IV, Springer, New York, pp. 409-418), instances arise in practice in which an estimation rule is to be accompanied by an estimate of its loss, which is unobservable. In the context of estimating the mean vector of a multi-normal distribution assuming a known population variance, Johnstone (In: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics, Vol. IV, Springer, New York, pp. 361-379) derived an estimator that dominates the unbiased estimator of the quadratic loss incurred by the James-Stein estimator. By applying the Stein's lemma, this note generalizes Johnstone's (In: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics, Vol. IV, Springer, New York, pp. 361-379) analysis to the setting of the unknown population variance. Computational evidence is provided about the risk magnitude of loss estimators associated with the James-Stein point estimator and its positive-part version. © 2002 Elsevier B.V. All rights reserved.