MINIMAX ESTIMATION OF THE GAUSSIAN PARAMETRIC REGRESSION

The paper considers the problem of estimating a d >= 2 dimensional mean vector of a multivariate normal distribution under quadratic loss. Let the observations be described by the equation Y = theta + sigma xi, (1) where. is a d-dimension vector of unknown parameters from some bounded set Theta subset of R-d, xi is a Gaussian random vector with zero mean and identity covariance matrix I-d, i.e. Law(xi)=N-d(0, I-d) and sigma is a known positive number. The problem is to construct a minimax estimator of the vector. from observations Y. As a measure of the accuracy of estimator (theta) over cap we select the quadratic risk defined as R(theta, (theta) over cap) := E-theta vertical bar theta - (theta) over cap vertical bar(2) , vertical bar x vertical bar(2) = Sigma(d)(j=1) x(j)(2), where E-theta is the expectation with respect to measure P-theta. We propose a modification of the James - Stein procedure of the form theta(*)(+) = (1 - c/vertical bar Y vertical bar)(+) Y where c > 0 is a special constant and a(+) = max(a, 0) is a positive part of a. This estimate allows one to derive an explicit upper bound for the quadratic risk and has a significantly smaller risk than the usual maximum likelihood estimator and the estimator theta(*) = (1 - c/vertical bar Y vertical bar)(+) Y for the dimensions d >= 2. We establish that the proposed procedure theta(*)(+) is minimax estimator for the vector theta. A numerical comparison of the quadratic risks of the considered procedures is given. In conclusion it is shown that the proposed minimax estimator theta(*)(+) is the best estimator in the mean square sense.