Statistical inference for models with multivariate t-distributed errors / A.K. Md. Ehsanes Saleh, M. Arashi, S.M.M. Tabatabaey.

Format:

Book

Author/Creator:

Saleh, A. K. Md. Ehsanes, author.

Arashi, M. (Mohammad), 1981- author.

Tabatabaey, S. M. M., author.

Contributor:

Wiley InterScience (Online service)

John Morgan Society Fund.

Language:

English

Subjects (All):

Regression analysis.

Multivariate analysis.

Physical Description:

1 online resource (xxiv, 248 pages)

Place of Publication:

Hoboken, New Jersey : Wiley, [2014]

System Details:

text file

Summary:

Uniquely presents systematic analytical results using Student's t-distributed errors in linear models, Statistical inference for Models with Multivariate t-Distributed Errors presents a wide array of applications for the analysis of multivariate observations and emphasizes the Students t-distribution method. The book illustrates the development of linear statistical models with applications to a variety of fields including mathematics, statistics, biostatistics, engineering, and the physical sciences. The book begins with a summary of the results under normal theory and proceeds to the statistical analysis of location models, simple regression, analysis of variance (ANOVA), parallelism, multiple regression, ridge regression, multivariate and simple multivariate linear models, and linear prediction. Providing a clear and balanced introduction to statistical inference, the book also features: A unique connection to normal distribution, Bayesian analysis, prediction problems, and Stein shrinkage estimation, Practical real-world examples that address linear regression models with non-normal errors, Plentiful end-of-chapter problems that enhance the applications for the analysis of multivariate observations, An up-to-date bibliography with the latest trends and advances to provide a collective resource for research, Statistical Inference for Models with Multivariate t-Distributed Errors is an excellent upper-undergraduate and graduate-level textbook for courses in multivariate analysis, regression, linear models, and Bayesian analysis. The book is also a useful resource for statistical practitioners who need solid methodology within mathematical and quantitative statistics. Book jacket.

Contents:

1 Introduction 1

1.1 Objective of the Book 1

1.2 Models under Consideration 3

1.2.1 Location Model 3

1.2.2 Simple Linear Model 4

1.2.3 ANOVA Model 5

1.2.4 Parallelism Model 6

1.2.5 Multiple Regression Model 7

1.2.6 Ridge Regression 8

1.2.7 Multivariate Model 8

1.2.8 Simple Multivariate Linear Model 9

1.3 Organization of the Book 9

1.4 Problems 10

2 Preliminaries 13

2.1 Normal Distribution 14

2.2 Chi-Square Distribution 14

2.3 Student's t-Distribution 16

2.4 F-Distribution 20

2.5 Multivariate Normal Distribution 22

2.6 Multivariate t-Distribution 23

2.6.1 Expected Values of Functions of M<sup>(p)</sub>_t, (η, σ²V<sub>p, γ₀)-Variables 25

2.6.2 Sampling Distribution of Quadratic Forms 26

2.6.3 Distribution of Linear Functions of t-Variables 32

2.7 Problems 34

3 Location Model 39

3.1 Model Specification 40

3.2 Unbiased Estimates of θ and σ² and Test of Hypothesis 40

3.3 Estimators 44

3.4 Bias and MSB Expressions of the Location Estimators 46

3.4.1 Analysis of the Estimators of Location Parameter 48

3.5 Various Estimates of Variance 56

3.5.1 Bias and MSB Expressions of the Variance Estimators 57

3.5.2 Analysis of the Estimators of the Variance Parameter 64

3.6 Problems 67

4 Simple Regression Model 69

4.1 Introduction 70

4.2 Estimation and Testing of η 70

4.2.1 Estimation of η 70

4.2.2 Test of Intercept Parameter 71

4.2.3 Estimators of β and θ 73

4.3 Properties of Intercept Parameter 74

4.3.1 Bias Expressions of the Estimators 74

4.3.2 MSE Expressions of the Estimators 75

4.4 Comparison 77

4.4.1 Optimum Level of Significance of θ<sub>n<sub>^PT 79

4.5 Numerical Illustration 80

4.6 Problems 85

5 Anova 87

5.1 Model Specification 88

5.2 Proposed Estimators and Testing 88

5.3 Bias, MSE, and Risk Expressions 93

5.4 Risk Analysis 95

5.4.1 Comparison of θ_n and θ_n 95

5.4.2 Comparison of θ^PT_n and θ _n (θ_n 96

5.4.3 Comparison of θ^S_n, θ _n, θ_n, and θ_n^PT 97

5.4.4 Comparison of θ^S_n and θ^S_n 98

5.5 Problems 100

6 Parallelism Model 101

6.1 Model Specification 101

6.2 Estimation of the Parameters and Test of Parallelism 103

6.2.1 Test of Parallelism 105

6.3 Bias, MSE, and Risk Expressions 109

6.3.1 Expressions of Bias, MSE Matrix, and Risks of β_n, Θ_n, β_n, and Θ_n 109

6.3.2 Expressions of Bias, MSE Matrix, and Risks of the PTEs of PTEs of β and Θ 110

6.3.3 Expressions of Bias, MSE Matrix, and Risks of the SSEs of β and Θ 111

6.3.4 Expressions of Bias. MSE Matrix, and Risks of the PRSEs of β and Θ 111

6.4 Risk Analysis 113

6.5 Problems 116

7 Multiple Regression Model 117

7.1 Model Specification 118

7.2 Shrinkage Estimators and Testing 118

7.3 Bias and Risk Expressions 122

7.3.1 Balanced Loss Function 122

7.3.2 Properties 123

7.4 Comparison 126

7.5 Problems 132

8 Ridge Regression 133

8.1 Model Specification 134

8.2 Proposed Estimators 135

8.3 Bias, MSE, and Risk Expressions 136

8.3.1 Biases of the Estimators 136

8.3.2 MSE Matrices and Risks of the Estimators 138

8.4 Performance of the Estimators 141

8.4.1 Comparison between β<sub>n(k), β^s_n(k), and β^s+_n (k) 142

8.4.2 Comparison between β_n(k) and β^PT_n (k) 144

8.4.3 Comparison between β_n(k) and β_n 144

8.4.4 Comparison between β_n(k) and β_n 145

8.4.5 Comparison between β^PT_n and β^PT_n(k) 148

8.4.6 Comparison between β^s_n and β^s_n(K) 151

8.4.7 Comparison between β^s+_n and β^s+_n(k) 154

8.5 Choice of Ridge Parameter 160

8.5.1 Real Example 160

8.5.2 Simulation 164

8.6 Problems 170

9 Multivariate Models 171

9.1 Location Model 172

9.2 Testing of Hypothesis and Several Estimators of Local Parameter 173

9.3 Bias, Quadratic Bias, MSE, and Risk Expressions 175

9.4 Risk Analysis of the Estimators 177

9.4.1 Comparison between Θ_N, Θ_N, and Θ^PT_N 177

9.4.2 Comparison between Θ_N, Θ_N, Θ^PT_N and Θ^S_N 179

9.4.3 Comparison between Θ_N, Θ^S_N, and Θ^S+_N 180

9.5 Simple Multivariate Linear Model 381

9.5.1 More Estimators for β and Θ 182

9.5.2 Bias, Quadratic Bias, and MSE Expressions 182

9.6 Problems 185

10 Bayesian Analysis 187

10.1 Introduction (Zellner's Model) 187

10.2 Conditional Bayesian Inference 189

10.3 Matrix Variate t-Distribution 191

10.4 Bayesian Analysis in Multivariate Regression Model 193

10.4.1 Properties of B and Φ 197

10.5 Problems 200

11 Linear Prediction Models 201

11.1 Model and Preliminaries 202

11.2 Distribution of SRV and RSS 203

11.3 Regression Model for Future Responses 205

11.4 Predictive Distributions of FRV and FRSS 206

11.4.1 Distribution of the FRV 207

11.4.2 Distribution of Future Residual Sum of Squares 212

11.5 An Illustration 212

11.6 Problems 213

12 Stein Estimation 215

12.1 Class of Estimators 216

12.1.1 Without Prior Information 216

12.1.2 Taking Prior Information 217

12.2 Itoliminaries and Some Theorems 219

12.3 Superiority Conditions 222

12.3.1 Without Taking Prior Information 222

12.3.2 Taking Prior Information 227

12.4 Problems 228.

Notes:

Includes bibliographical references and index.

Electronic reproduction. Hoboken, N.J. Available via World Wide Web.

Description based on online resource; title from digital title page (viewed on October 20, 2014).

Local Notes:

Acquired for the Penn Libraries with assistance from the John Morgan Society Fund.

Other Format:

Print version: Saleh, A. K. Md. Ehsanes, author Statistical inference for models with multivariate t-distributed errors

ISBN:

9781118853931

1118853938

9781118853924

111885392X

9781118853962

1118853962

Publisher Number:

99960382822

Access Restriction:

Restricted for use by site license.