Shrinkage estimation of a mean vector in a two-sample problem
Let X 1 X 2 be two p-variate (p ≥ 3) normal random observations and U be a chi-square random variable, where X 1 ∼ N p (μ1,σ 2Σ1)X 2 ∼ N p (μ2,σ 2Σ2), and Assume that X 1 X 2, and U are mutually independent. We wish to estimate the mean vector μ1 based on X 1 X 2, and U when Σ1 is known, but σ 2 and Σ2 are unknown. Two families of minimax estimators are proposed in spite of the fact that independent estimates of σ 2Σ2 are unavailable.