Statistical Inference for Models with Multivariate t-Distributed Errors

A. K. Md. Ehsanes Saleh, PhD, is Professor Emeritus and Distinguished Research Professor in the School of Mathematics and Statistics at Carleton University, Canada. He has published well-over 200 journal articles, and his research interests include nonparametric statistics, order statistics, and robust estimation.  Dr. Saleh is a Fellow of the Institute of Mathematical Statistics, the American Statistical Association, and the Bangladesh Academy of Sciences. M. Arashi, PhD, is Associate Professor in the Department of Statistics at Shahrood University of Technology, Iran.  The recipient of the Award for Teaching Excellence from Shahrood University in 2013, his research interests include shrinkage estimation, distribution theory, and multivariate analysis.  S. M. M. Tabatabaey, PhD, is Associate Professor in the Department of Statistics at Ferdowski University of Mashhad, Iran. The author of over fifteen journal articles, he is also a member of the Institute of Mathematical Statistics and the Iranian Statistical Society.

List of Figures xv

List of Tables xvii

Preface xix

Glossary xxi

List of Symbols xxiii

1 Introduction 1

1.1 Objective of the Book 1

1.2 Models Under Consideration 3

2 Preliminaries 7

2.1 Normal distribution 8

2.2 Chisquare distribution 8

2.3 Student’s t distributions 10

2.4 F distribution 14

2.5 Multivariate Normal distribution 16

2.6 Multivariate t distribution 17

2.7 Problems 28

3 Location Model 31

3.1 Model Specification 32

3.2 Unbiased Estimates of _ and _2 and test of hypothesis 32

3.3 Estimators 36

3.4 Bias and MSE Expressions of the Location Estimators 38

3.5 Various Estimates of Variance 48

3.6 Problems 60

4 Simple Regression Model 61

4.1 Introduction 62

4.2 Estimation and Testing of _ 62

4.3 Properties of Intercept Parameter 66

4.4 Comparison 69

4.5 Numerical Illustration 72

4.6 Problems 77

5 ANOVA 79

5.1 Model Specification 80

5.2 Proposed Estimators and Testing 80

5.3 Bias, MSE and Risk Expressions 85

5.4 Risk Analysis 87

5.5 Problems 93

6 Parallelism Model 95

6.1 Model Specification 96

6.2 Estimation of the Parameters and Test of Parallelism 97

6.3 Bias, MSE, and Risk Expressions 103

6.4 Risk Analysis 106

6.5 Problems 110

7 Multiple Regression Model 111

7.1 Model Specification 112

7.2 Shrinkage Estimators and Testing 112

7.3 Bias and Risk Expressions 116

7.4 Comparison 120

7.5 Problems 126

8 Ridge Regression 127

8.1 Model Specification 128

8.2 Proposed Estimators 129

8.3 Bias, MSE, and Risk Expressions 130

8.4 Performance of the Estimators 135

8.5 Choice of Ridge Parameter 153

8.6 Problems 164

9 Multivariate Models 165

9.1 Location Model 166

9.2 Testing of Hypothesis and Several Estimators of Local Parameter 167

9.3 Bias, Quadratic Bias, MSE, and Risk Expressions 169

9.4 Risk Analysis of the Estimators 171

9.5 Simple Multivariate Linear Model 175

9.6 Problems 180

10 Bayesian Analysis 181

10.1 Introduction (Zellner’s Model) 181

10.2 Conditional Bayesian Inference 183

10.3 Matrix Variate t Distribution 185

10.4 Bayesian Analysis in Multivariate Regression Model 187

10.5 Problems 194

11 Linear Prediction Models 195

11.1 Model & Preliminaries 196

11.2 Distribution of SRV and RSS 197

11.3 Regression Model for Future Responses 199

11.4 Predictive Distributions of FRV and FRSS 200

11.5 An Illustration 206

11.6 Problems 208

12 Stein Estimation 209

12.1 Class of Estimators 210

12.2 Preliminaries and Some Theorems 213

12.3 Superiority Conditions 216

12.4 Problems 223

References 225

Subject Index 243