## Electronic Theses and Dissertations

2012

Dissertation

Ph.D.

#### Department

Mathematics and Statistics

Ahmed, Syed Ejaz (Economics, Mathematics, and Statistics)

Statistics.

CC BY-NC-ND 4.0

#### Abstract

The objective of this dissertation is to study properties of improved estimators of the parameters of interest in two different multivariate regression models, analogous to the fixed-X and random-X scenarios of multiple regression and compare the performance of these estimators with the usual least square estimator. In general, we study restricted versions of the multivariate regression problem based upon constraining the relationship between Y and X in some way where they may be known or unknown to the researcher prior to statistical analysis. Chapter two contains a study of the properties of improved estimation strategies for the parameters of interest in a capital asset pricing model under a general linear constraint. Asymptotic results of the suggested estimators include derivation of asymptotic bias, asymptotic mean square error, and asymptotic distributional risk. The asymptotic results demonstrate the superiority of the suggested estimation technique. A simulation study is conducted to assess the performance of the suggested estimators for large samples. Both simulation study and data example corroborate with the theoretical result. In Chapter three, we consider a multivariate multiple regression model when X is a fixed matrix. Here, we propose shrinkage and preliminary test estimation strategies for the matrix of regression parameters in the presence of a natural linear constraint. We examine the relative performances of the suggested estimators under the candidate subspace based on a quadratic risk function and the results are shown. A simulation study is conducted to compare the performance of the suggested estimators and two data examples are also presented. Our analytical and numerical results show that the suggested estimators perform better than the unrestricted estimator under the candidate subspace. In Chapter four, we consider a multivariate reduced rank regression model when X is random and we propose preliminary test and shrinkage estimation strategies. We investigate the asymptotic properties of the shrinkage and pretest estimators under a quadratic loss function and compare the performance of the suggested estimators under the candidate subspace and beyond. The methods are applied on a real data set for illustrative purposes and a simulation study is also presented.

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