Statistical Inference for Models with Multivariate t-Distributed Errors

This book summarizes the results of various models under normal theory with a brief review of the literature. __Statistical Inference for Models with Multivariate t-Distributed Errors__: * Includes a wide array of applications for the analysis of multivariate observations * Emphasizes the development of linear statistical models with applications to engineering, the physical sciences, and mathematics * Contains an up-to-date bibliography featuring the latest trends and advances in the field to provide a collective source for research on the topic * Addresses linear regression models with non-normal errors with practical real-world examples * Uniquely addresses regression models in Student's __t__-distributed errors and__t__-models * Supplemented with an Instructor's Solutions Manual, which is available via written request by the Publisher Cover 1 Title Page 5 Copyright Page 6 CONTENTS 11 List of Figures 17 List of Tables 19 Preface 21 Glossary 23 List of Symbols 25 1 Introduction 27 1.1 Objective of the Book 27 1.2 Models under Consideration 29 1.2.1 Location Model 29 1.2.2 Simple Linear Model 30 1.2.3 ANOVA Model 31 1.2.4 Parallelism Model 32 1.2.5 Multiple Regression Model 33 1.2.6 Ridge Regression 34 1.2.7 Multivariate Model 34 1.2.8 Simple Multivariate Linear Model 35 1.3 Organization of the Book 35 1.4 Problems 36 2 Preliminaries 39 2.1 Normal Distribution 40 2.2 Chi-Square Distribution 40 2.3 Student's t-Distribution 42 2.4 F-Distribution 46 2.5 Multivariate Normal Distribution 48 2.6 Multivariate t-Distribution 49 2.6.1 Expected Values of Functions of M_t^(p)(η , σ^2V_p, γo) - Variables 51 2.6.2 Sampling Distribution of Quadratic Forms 52 2.6.3 Distribution of Linear Functions of t-Variables 58 2.7 Problems 60 3 Location Model 65 3.1 Model Specification 66 3.2 Unbiased Estimates of θ and σ^2 and Test of Hypothesis 66 3.3 Estimators 70 3.4 Bias and MSE Expressions of the Location Estimators 72 3.4.1 Analysis of the Estimators of Location Parameter 74 3.5 Various Estimates of Variance 82 3.5.1 Bias and MSE Expressions of the Variance Estimators 83 3.5.2 Analysis of the Estimators of the Variance Parameter 90 3.6 Problems 93 4 Simple Regression Model 95 4.1 Introduction 96 4.2 Estimation and Testing of η 96 4.2.1 Estimation of η 96 4.2.2 Test of Intercept Parameter 97 4.2.3 Estimators of β and θ 99 4.3 Properties of Intercept Parameter 100 4.3.1 Bias Expressions of the Estimators 100 4.3.2 MSE Expressions of the Estimators 101 4.4 Comparison 103 4.4.1 Optimum Level of Significance of θ_n^PT 105 4.5 Numerical Illustration 106 4.6 Problems 111 5 ANOVA 113 5.1 Model Specification 114 5.2 Proposed Estimators and Testing 114 5.3 Bias, MSE, and Risk Expressions 119 5.4 Risk Analysis 121 5.4.1 Comparison of θ_n and θ_n 121 5.4.2 Comparison of θ_n_PT and θ_n(θ_n) 122 5.4.3 Comparison of θ_n^S, θ_n , θn, and θ_n^PT 123 5.4.4 Comparison of θ_n^S and θ_n^S+ 124 5.5 Problems 126 6 Parallelism Model 127 6.1 Model Specification 127 6.2 Estimation of the Parameters and Test of Parallelism 129 6.2.1 Test of Parallelism 131 6.3 Bias, MSE, and Risk Expressions 135 6.3.1 Expressions of Bias, MSE Matrix, and Risks of β_n, Θ_n, β_n, and Θ_n 135 6.3.2 Expressions of Bias, MSE Matrix, and Risks of the PTEs of β and Θ 136 6.3.3 Expressions of Bias, MSE Matrix, and Risks of the SSEs of β and Θ 137 6.3.4 Expressions of Bias, MSE Matrix, and Risks of the PRSEs of β and Θ 137 6.4 Risk Analysis 139 6.5 Problems 142 7 Multiple Regression Model 143 7.1 Model Specification 144 7.2 Shrinkage Estimators and Testing 144 7.3 Bias and Risk Expressions 148 7.3.1 Balanced Loss Function 148 7.3.2 Properties 149 7.4 Comparison 152 7.5 Problems 158 8 Ridge Regression 159 8.1 Model Specification 160 8.2 Proposed Estimators 161 8.3 Bias, MSE, and Risk Expressions 162 8.3.1 Biases of the Estimators 162 8.3.2 MSE Matrices and Risks of the Estimators 164 8.4 Performance of the Estimators 167 8.4.1 Comparison between β_n(k), β_n^S(k), and β_n^S+(k) 168 8.4.2 Comparison between β_n (k) and β_n^PT (k) 170 8.4.3 Comparison between β_n(k) and β_n 170 8.4.4 Comparison between β_n (k) and β_n 171 8.4.5 Comparison between (β_n^PT and β_n^PT (k) 174 8.4.6 Comparison between β_n^S and β_n^S (k) 177 8.4.7 Comparison between β_n^S+ and β_n^S+ (k) 180 8.5 Choice of Ridge Parameter 186 8.5.1 Real Example 186 8.5.2 Simulation 190 8.6 Problems 196 9 Multivariate Models 197 9.1 Location Model 198 9.2 Testing of Hypothesis and Several Estimators of Local Parameter 199 9.3 Bias, Quadratic Bias, MSE, and Risk Expressions 201 9.4 Risk Analysis of the Estimators 203 9.4.1 Comparison between θ_N, θ_N, and θ_N^PT 203 9.4.2 Comparison between θ_N, θ_N, θ_N^PT, and θ_N^S 205 9.4.3 Comparison between θ_NN, θ_N^S, and θ_N^S+ 206 9.5 Simple Multivariate Linear Model 207 9.5.1 More Estimators for β and θ 208 9.5.2 Bias, Quadratic Bias, and MSE Expressions 208 9.6 Problems 211 10 Bayesian Analysis 213 10.1 Introduction (Zellner's Model) 213 10.2 Conditional Bayesian Inference 215 10.3 Matrix Variate t-Distribution 217 10.4 Bayesian Analysis in Multivariate Regression Model 219 10.4.1 Properties of B and Φ 223 10.5 Problems 226 11 Linear Prediction Models 227 11.1 Model and Preliminaries 228 11.2 Distribution of SRV and RSS 229 11.3 Regression Model for Future Responses 231 11.4 Predictive Distributions of FRV and FRSS 232 11.4.1 Distribution of the FRV 233 11.4.2 Distribution of Future Residual Sum of Squares 238 11.5 An Illustration 238 11.6 Problems 239 12 Stein Estimation 241 12.1 Class of Estimators 242 12.1.1 Without Prior Information 242 12.1.2 Taking Prior Information 243 12.2 Preliminaries and Some Theorems 245 12.3 Superiority Conditions 248 12.3.1 Without Taking Prior Information 248 12.3.2 Taking Prior Information 253 12.4 Problems 254 References 255 Author Index 267 Subject Index 271 EULA 275 "This book summarizes the results of various models under normal theory with a brief review of the literature. Statistical Inference for Models with Multivariate t-Distributed Errors: Includes a wide array of applications for the analysis of multivariate observations Emphasizes the development of linear statistical models with applications to engineering, the physical sciences, and mathematics Contains an up-to-date bibliography featuring the latest trends and advances in the field to provide a collective source for research on the topic Addresses linear regression models with non-normal errors with practical real-world examples Uniquely addresses regression models in Student's t-distributed errors and t-models Supplemented with an Instructor's Solutions Manual, which is available via written request by the Publisher "-- Provided by publisher This book summarizes the results of various models under normal theory with a brief review of the literature. Statistical Inference for Models with Multivariate t-Distributed Errors : Includes a wide array of applications for the analysis of multivariate observations Emphasizes the development of linear statistical models with applications to engineering, the physical sciences, and mathematics Contains an up-to-date bibliography featuring the latest trends and advances in the field to provide a collective source for research on the topic Addresses linear regression models with non-normal errors with practical real-world examples Uniquely addresses regression models in Student's t -distributed errors and t -models Supplemented with an Instructor's Solutions Manual, which is available via written request by the Publisher