Minimaxity and admissibility of matrix shrinkage estimators

This master thesis refers to the determination of certain choice criteria for minimaxes and admissibles shrinkage estimators. Adopting the coordinate-free approach, the problem of estimating the mean of a multidimensional normal distribution over a n-dimensional real vector space, is being studied. In section 1 we study the problem of estimating the mean of a multidimensional normal distribution when the variance is known up to a multiplicative factor. The risk evaluation allows us to present sufficient conditions of domination over the least square estimator (lse). In section 2, we originally study the subject of estimating non-physical parameters in exponential distribution families. An interesting special case leads to the problem described in section 1, while a necessary condition for the admissibility of shrinkage estimators can be concluded from the general result. In section 3, we offer a necessary and sufficient condition of domination over lse. In section 4 we succeed in improving the estimators described in the previous sections. We initially strengthen the shrinkage through the shrinkage endomorphism and afterwards we modify or restrain the shrinkage function. At the end of this section, we estimate using simulation methods the upper bound of the gain that can be obtained by a shrinkage estimator in relation to the sleuths allows a compare of the estimators gain with that specific bound