Unbiased equivariant estimation of a common normal mean vector with one observation from each population

Let X1 be a random observation from a p-variate normal population with mean vector [theta] and covariance matrix proportional to identity matrix, Np([theta], [sigma]21Ip). In addition to X1, there is another observation X2 from Np([theta], [sigma]22Ip). In this note, an unbiased estimator which combines both X1 and X2 is developed and its risk behavior is studied. Then, assuming that [sigma]21 is known, a motivation for the best shrinkage estimator in a class of estimators that shrink X1 toward X2 is given. It is shown that such shrinkage estimators are unbiased and location equivariant. Also, for a shrinkage estimator from this class the risk improvements over X1 and the one that shrinks toward the origin are studied.