Minimax estimation under generalized quadratic loss

An admissible minimax estimate is derived for the following statistical decision problem. Let z1 = x + u + v1 and z2 = x + v2, where x ?? N[o, Q], v1 ?? N[o, R1], v2 ?? N[o, R2], and u ?? En. A statistician observes the random vector z1 and seeks a minimax estimate ??(z1) for the sum (x + u), under the generalized quadratic loss function defined by L(??, u) = [?? - (x + u)]´ C[?? - (x + u)] - u´du. Nature observes the random vector z2 and controls the value of the vector u, which she may make dependent on the observed value of z2. Both parties know the covariance matrices of the indicated normal random vectors, which are assumed to be statistically independent. The minimax decision rule is shown to be linear, and nature´s optimum choice of u is shown to be u = Pz2, where the matrix P is determined by the solution to a certain nonlinear matrix equation.