Pitman's Measure of Closeness on Generalized Class of Biased Estimator

In this dissertation, part of main results on PMC criterion in linear model is briefly summarized. While Pitman closeness may be used as a criterion in estimation it is best to simply view the measure as an aid in understanding the intricacies of conditional risk, when an estimator is closer and when it is not. The basic linear model and some definitions are presented, then PMC superiority for a class of biased estimators, such as a kind of ridge estimate, James-Stein type estimate and the combined principal components estimate will be discussed. One objective of this dissertation is to develop such methodology for the case where the squared statistical distance between the estimator and parameter is used as the loss function; the estimators to be compared are linear combinations of a common statistics and the statistic has a multivariate normal distribution. To demonstrate the utility of the methodology, it will be applied to the problem of comparing biased r - k class estimators to unbiased least-squares estimator of regression coefficients. After the theory has been developed for the r - k class estimators, a data set from the literature will be used to illustrate the calculation and comparative techniques.