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 -0+а^, (1) where 0 is a d-dimension vector of unknown parameters from some bounded set 0c8 , £ is a Gaussian random vector with zero mean and identity covariance matrix Id, i.e. .Law(£)=Nd(0, Id) and a is a known positive number. The problem is to construct a minimax estimator of the vector 0 from observations Y. As a measure of the accuracy of estimator 0 we select the quadratic risk defined as R(0,0):- E ₀- 0| , |x| -]Tx , j-1 where E ₀ is the expectation with respect to measure P ₀ . We propose a modification of the James - Stein procedure of the form 0+-I1 I Y |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 0*-|1 -.£- |Y |Y|, for the dimensions d > 2. We establish that the proposed procedure 0+ is minimax estimator for the vector 0. A numerical comparison of the quadratic risks of the considered procedures is given. In conclusion it is shown that the proposed minimax estimator 0+ is the best estimator in the mean square sense.

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