On a minimax parameter estimation problem with a compact parameter space

A minimax estimate for the mean of an arbitrary random vector with known covariance is derived, using a standard quadratic loss function, subject to the restriction that the decision rule be linear and that the unknown mean lie in a known hyperellipsoidal subset of En. This linear minimax estimate is derived using the method of least favorable prior distributions. It is shown that there is a least favorable prior distribution which is defined by a discrete probability measure with support which lies entirely on the boundary of the hyperellipsoidal parameter space. This discrete distribution is supported by 2k points, which receive equal probability, and which are determined by the solution to a certain nonlinear algebraic matrix equation. The exponent k is equal to the rank of the covariance matrix of this least favorable prior distribution.