Signal estimation under restrictions: Minimaxity and shrinkag

This dissertation discusses several aspects of signal estimation under restrictions on the parameter space. The second chapter studies the problem of estimating the mean of a multivariate normal distribution when the parameter space can be written as a direct sum of subspaces, for instance in analysis of variance situations. Stein-type estimators for this situation are presented and their asymptotic behavior analyzed. In chapter three, we study the special case of shrinkage estimation in linear regression. Different shrinkage schemes are presented and their performance evaluated on real data of near-infrared measurements on food, where extreme collinearity is often a problem. Chapter four deals with minimax estimation over linear estimators in regression under certain restrictions on the parameter space. 'Oracle' estimators achieving Pinsker's minimax bound when the size of the parameter space is known are deduced and shown to be superior to ridge regression. It is argued that an adaptive version of this estimator where the size of the parameter space is estimated from the data is the natural 'Stein' estimator in regression problem. The minimax estimator is related to 'Speckman' splines in nonparametric regression. The last chapter discusses construction of confidence sets for a class of linear smoothers that includes minimax regression, ridge regression and principal components regression. A simulation study compares confidence sets centered at these estimators using asymptotic theory and bootstrap.