Shrinkage, GMANOVA, control variates and their applications

Ming Tan, Purdue University

Abstract

The problem of finding classes of estimators which improve upon the usual (e.g. ML, LS) estimator of the parameter matrix in the GMANOVA model is considered under both matrix quadratic loss an a general total squared error loss. Unbiased estimators of risk differences for certain classes of estimators are obtained via combining integration-by-parts methods for normal and Wishart distributions, thereby extending results in Gleser (1986). Then, classes of improved estimators for the parameter matrix in GMANOVA model are given. Further, a comparison of two related classes of estimators is made, and an analytic proof for risk dominance is obtained in a special case. As an application of the above results, the common mean estimation problem is considered. Variance reduction is an important issue in using Monte Carlo simulation to analyze the performance of large-scale systems. The control variates method has widely been used as a means for improving efficiency of the simulation. It is demonstrated in this thesis that further variance reduction can be achieved by using Stein shrinkage estimation. That is, improved estimators of the mean response matrix can be obtained by applying the results concerning the improved estimators in GMANOVA to the meta-model of the control variates. This problem is treated from several approaches in order to obtain further variance reduction, especially for the one population case. Finally, reporting a confidence interval for a mean (or mean vector) $\mu$ is of more practical interest than simply reporting a point estimator. An improved confidence interval for the mean response in Monte Carlo simulation is obtained by recentering the usual confidence interval at a particular shrinkage estimator. Numerical evidence is given to support the improved performance of this kind of interval.

Degree

Ph.D.

Advisors

Gleser, Purdue University.

Subject Area

Statistics

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