Coverage probability of confidence sets centered at James-Stein estimators and its positive part

Confidence sets and coverage probability are the main topics considered in this thesis. Definitions and properties of point estimation theory and decision theory are discussed; the estimate of the mean vector for a random vector of observations following a multivariate normal distribution is found. The popular method of maximum likelihood estimation is used to find the estimate and then an enhancement with respect to the mean square error can be done with a shrinkage estimator known as the James-Stein estimator, which is obtained by contracting the usual mean by a factor; further improvement can be done by considering the positive part of the latter estimator.;Numerical calculations aimed toward pointing out the improvement of the coverage probability of the confidence set centered at the positive part are shown. For that purpose programs created in MATLAB and R are used.;Hwang and Casella proved that the confidence set centered at the positive part of the James-Stein estimator has a higher coverage probability than the confidence set centered at the maximum likelihood estimator (usual mean). Motivating by this result, simple formulas for the coverage probability of the confidence set centered at the positive part of the James-Stein estimator will be deduced from an analytical and geometrical point of view. There were no exact formulas before for the coverage probability of confidence sets centered at James-Stein estimators and its positive part.