Shrinkage estimators for the location parameter of an elliptically symmetric distribution in a general linear model

The subject of this master thesis is shrinkage estimators for the location parameter of an elliptically symmetric distribution in a general linear model. The theory regarding these type of estimators has been introduced by James and Stein (1961), while Stein (1956) proved that the usual least square estimator (lse) is not admissible when the dimension of the parameter space is greater than 2. Ever since, manifold generalizations have taken place that refer to domination over lse and performance of this class of estimators. This project covers part of this theory from 1985 to 1994 and is divided in 5 sections. In section 1, we study elliptically symmetric distributions, give characterizations and prove many of their interesting properties. In section 2, we present a class of shrinkage estimators for the location parameter of a multidimensional normal distribution. We offer a condition of domination over lse, submitting any analytical properties of the shrinkage function. In section 3, we generalize our previous results under elliptical symmetry for a general quadratic loss. Section 4 refers to shrinkage estimators for the location parameter of an elliptically symmetric distribution where the shrinkage function is replaced by a differentiable vector function. Finally, in section 5 we present two applications of shrinkage estimators (particularly Steins estimator) on rainfall data in co-operation with IACM-FORTH