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# Theory of Ridge Regression Estimation with Applications

ISBN: 978-1-118-64450-8 January 2019 384 Pages

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## Description

A guide to the systematic analytical results for ridge, LASSO, preliminary test, and Stein-type estimators with applications

Theory of Ridge Regression Estimation with Applications offers a comprehensive guide to the theory and methods of estimation. Ridge regression and LASSO are at the center of all penalty estimators in a range of standard models that are used in many applied statistical analyses. Written by noted experts in the field, the book contains a thorough introduction to penalty and shrinkage estimation and explores the role that ridge, LASSO, and logistic regression play in the computer intensive area of neural network and big data analysis.

Designed to be accessible, the book presents detailed coverage of the basic terminology related to various models such as the location and simple linear models, normal and rank theory-based ridge, LASSO, preliminary test and Stein-type estimators. The authors also include problem sets to enhance learning. This book is a volume in the Wiley Series in Probability and Statistics series that provides essential and invaluable reading for all statisticians. This important resource:

• Offers theoretical coverage and computer-intensive applications of the procedures presented
• Contains solutions and alternate methods for prediction accuracy and selecting model procedures
• Presents the first book to focus on ridge regression and unifies past research with current methodology
• Uses R throughout the text and includes a companion website containing convenient data sets

Written for graduate students, practitioners, and researchers in various fields of science, Theory of Ridge Regression Estimation with Applications is an authoritative guide to the theory and methodology of statistical estimation.

List of Figures xvii

List of Tables xxi

Preface xxvii

Abbreviations and Acronyms xxxi

List of Symbols xxxiii

1 Introduction to Ridge Regression 1

1.1 Introduction 1

1.1.1 Multicollinearity Problem 3

1.2 Ridge Regression Estimator: Ridge Notion 5

1.3 LSE vs. RRE 6

1.4 Estimation of Ridge Parameter 7

1.5 Preliminary Test and Stein-Type Ridge Estimators 8

1.6 High-Dimensional Setting 9

1.7 Notes and References 11

1.8 Organization of the Book 12

2 Location and Simple Linear Models 15

2.1 Introduction 15

2.2 Location Model 16

2.2.1 Location Model: Estimation 16

2.2.2 Shrinkage Estimation of Location 17

2.2.3 Ridge Regression–Type Estimation of Location Parameter 18

2.2.4 LASSO for Location Parameter 18

2.2.5 Bias and MSE Expression for the LASSO of Location Parameter 19

2.2.6 Preliminary Test Estimator, Bias, and MSE 23

2.2.7 Stein-Type Estimation of Location Parameter 24

2.2.8 Comparison of LSE, PTE, Ridge, SE, and LASSO 24

2.3 Simple Linear Model 26

2.3.1 Estimation of the Intercept and Slope Parameters 26

2.3.2 Test for Slope Parameter 27

2.3.3 PTE of the Intercept and Slope Parameters 27

2.3.4 Comparison of Bias and MSE Functions 29

2.3.5 Alternative PTE 31

2.3.6 Optimum Level of Significance of Preliminary Test 33

2.3.7 Ridge-Type Estimation of Intercept and Slope 34

2.3.7.1 Bias and MSE Expressions 35

2.3.8 LASSO Estimation of Intercept and Slope 36

2.4 Summary and Concluding Remarks 39

3 ANOVA Model 43

3.1 Introduction 43

3.2 Model, Estimation, and Tests 44

3.2.1 Estimation of Treatment Effects 45

3.2.2 Test of Significance 45

3.2.3 Penalty Estimators 46

3.2.4 Preliminary Test and Stein-Type Estimators 47

3.3 Bias and Weighted L2 Risks of Estimators 48

3.3.1 Hard Threshold Estimator (Subset Selection Rule) 48

3.3.2 LASSO Estimator 49

3.3.3 Ridge Regression Estimator 51

3.4 Comparison of Estimators 52

3.4.1 Comparison of LSE with RLSE 52

3.4.2 Comparison of LSE with PTE 52

3.4.3 Comparison of LSE with SE and PRSE 53

3.4.4 Comparison of LSE and RLSE with RRE 54

3.4.5 Comparison of RRE with PTE, SE, and PRSE 56

3.4.5.1 Comparison Between 𝜽n^RR (kopt) and 𝜽n^PT (𝛼) 56

3.4.5.2 Comparison Between 𝜽n^RR (kopt) and 𝜽n^ s 56

3.4.5.3 Comparison of 𝜽n^RR (kopt) with 𝜽n^S+ 57

3.4.6 Comparison of LASSO with LSE and RLSE 58

3.4.7 Comparison of LASSO with PTE, SE, and PRSE 59

3.4.8 Comparison of LASSO with RRE 60

3.5 Application 60

3.6 Efficiency in Terms of Unweighted L2 Risk 63

3.7 Summary and Concluding Remarks 72

3A. Appendix 74

4 Seemingly Unrelated Simple Linear Models 79

4.1 Model, Estimation, and Test of Hypothesis 79

4.1.1 LSE of 𝜃 and 𝛽 80

4.1.2 Penalty Estimation of 𝛽 and 𝜃 80

4.1.3 PTE and Stein-Type Estimators of 𝛽 and 𝜃 81

4.2 Bias and MSE Expressions of the Estimators 82

4.3 Comparison of Estimators 86

4.3.1 Comparison of LSE with RLSE 86

4.3.2 Comparison of LSE with PTE 86

4.3.3 Comparison of LSE with SE and PRSE 87

4.3.4 Comparison of LSE and RLSE with RRE 87

4.3.5 Comparison of RRE with PTE, SE, and PRSE 89

4.3.5.1 Comparison Between 𝜽n^RR (kopt) and 𝜽n^PT 89

4.3.5.2 Comparison Between 𝜽n^RR (kopt) and 𝜽n^S 89

4.3.5.3 Comparison of 𝜽n^RR (kopt) with 𝜽n^S+ 90

4.3.6 Comparison of LASSO with RRE 90

4.3.7 Comparison of LASSO with LSE and RLSE 92

4.3.8 Comparison of LASSO with PTE, SE, and PRSE 92

4.4 Efficiency in Terms of Unweighted L2 Risk 93

4.4.1 Efficiency for 𝜷 94

4.4.2 Efficiency for 𝜽 95

4.5 Summary and Concluding Remarks 96

5 Multiple Linear Regression Models 109

5.1 Introduction 109

5.2 Linear Model and the Estimators 110

5.2.1 Penalty Estimators 111

5.2.2 Shrinkage Estimators 113

5.3 Bias and Weighted L2 Risks of Estimators 114

5.3.1 Hard Threshold Estimator 114

5.3.2 Modified LASSO 116

5.3.3 Multivariate Normal Decision Theory and Oracles for Diagonal Linear Projection 117

5.3.4 Ridge Regression Estimator 119

5.3.5 Shrinkage Estimators 119

5.4 Comparison of Estimators 120

5.4.1 Comparison of LSE with RLSE 120

5.4.2 Comparison of LSE with PTE 121

5.4.3 Comparison of LSE with SE and PRSE 121

5.4.4 Comparison of LSE and RLSE with RRE 122

5.4.5 Comparison of RRE with PTE, SE, and PRSE 123

5.4.5.1 Comparison Between 𝜽n^RR (kopt) and 𝜽n^PT(𝛼) 123

5.4.5.2 Comparison Between 𝜽n^RR (kopt) and 𝜽n^S 124

5.4.5.3 Comparison of 𝜽n^RR (kopt) with 𝜽n^S+ 124

5.4.6 Comparison of MLASSO with LSE and RLSE 125

5.4.7 Comparison of MLASSO with PTE, SE, and PRSE 126

5.4.8 Comparison of MLASSO with RRE 127

5.5 Efficiency in Terms of Unweighted L2 Risk 127

5.6 Summary and Concluding Remarks 129

6 Ridge Regression in Theory and Applications 143

6.1 Multiple Linear Model Specification 143

6.1.1 Estimation of Regression Parameters 143

6.1.2 Test of Hypothesis for the Coefficients Vector 145

6.2 Ridge Regression Estimators (RREs) 146

6.3 Bias, MSE, and L2 Risk of Ridge Regression Estimator 147

6.4 Determination of the Tuning Parameters 151

6.5 Ridge Trace 151

6.6 Degrees of Freedom of RRE 154

6.7 Generalized Ridge Regression Estimators 155

6.8 LASSO and Adaptive Ridge Regression Estimators 156

6.9 Optimization Algorithm 158

6.9.1 Prostate Cancer Data 160

6.10 Estimation of Regression Parameters for Low-Dimensional Models 161

6.10.1 BLUE and Ridge Regression Estimators 161

6.10.2 Bias and L2-risk Expressions of Estimators 162

6.10.3 Comparison of the Estimators 165

6.10.4 Asymptotic Results of RRE 166

6.11 Summary and Concluding Remarks 168

7 Partially Linear Regression Models 171

7.1 Introduction 171

7.2 Partial Linear Model and Estimation 172

7.3 Ridge Estimators of Regression Parameter 174

7.4 Biases and L2 Risks of Shrinkage Estimators 177

7.5 Numerical Analysis 178

7.5.1 Example: Housing Prices Data 182

7.6 High-Dimensional PLM 188

7.6.1 Example: Riboflavin Data 192

7.7 Summary and Concluding Remarks 193

8 Logistic Regression Model 197

8.1 Introduction 197

8.1.1 Penalty Estimators 199

8.1.2 Shrinkage Estimators 200

8.1.3 Results on MLASSO 201

8.1.4 Results on PTE and Stein-Type Estimators 202

8.1.5 Results on Penalty Estimators 204

8.2 Asymptotic Distributional L2 Risk Efficiency Expressions of the Estimators 204

8.2.1 MLASSO vs. MLE 205

8.2.2 MLASSO vs. RMLE 206

8.2.3 Comparison of MLASSO vs. PTE 206

8.2.4 PT and MLE 207

8.2.5 Comparison of MLASSO vs. SE 208

8.2.6 Comparison of MLASSO vs. PRSE 208

8.2.7 RRE vs. MLE 209

8.2.7.1 RRE vs. RMLE 209

8.2.8 Comparison of RRE vs. PTE 211

8.2.9 Comparison of RRE vs. SE 211

8.2.10 Comparison of RRE vs. PRSE 212

8.2.11 PTE vs. SE and PRSE 212

8.2.12 Numerical Comparison Among the Estimators 213

8.3 Summary and Concluding Remarks 213

9 Regression Models with Autoregressive Errors 221

9.1 Introduction 221

9.1.1 Penalty Estimators 223

9.1.2 Shrinkage Estimators 224

9.1.2.1 Preliminary Test Estimator 224

9.1.2.2 Stein-Type and Positive-Rule Stein-Type Estimators 225

9.1.3 Results on Penalty Estimators 225

9.1.4 Results on PTE and Stein-Type Estimators 226

9.1.5 Results on Penalty Estimators 229

9.2 Asymptotic Distributional L2-risk Efficiency Comparison 230

9.2.1 Comparison of GLSE with RGLSE 230

9.2.2 Comparison of GLSE with PTE 231

9.2.3 Comparison of LSE with SE and PRSE 231

9.2.4 Comparison of LSE and RLSE with RRE 232

9.2.5 Comparison of RRE with PTE, SE and PRSE 233

9.2.5.1 Comparison Between 𝜷n^GRR(kopt) and 𝜷n^G(PT)233

9.2.5.2 Comparison Between 𝜷n^GRR(kopt) and 𝜷n^G(S) 234

9.2.5.3 Comparison of 𝜷n^GRR(kopt) with 𝜷n^G(S+) 234

9.2.6 Comparison of MLASSO with GLSE and RGLSE 235

9.2.7 Comparison of MLASSO with PTE, SE, and PRSE 236

9.2.8 Comparison of MLASSO with RRE 236

9.3 Example: Sea Level Rise at KeyWest, Florida 237

9.3.1 Estimation of the Model Parameters 237

9.3.1.1 Testing for Multicollinearity 237

9.3.1.2 Testing for Autoregressive Process 238

9.3.1.3 Estimation of Ridge Parameter k 239

9.3.2 Relative Efficiency 240

9.3.2.1 Relative Efficiency (REff) 240

9.3.2.2 Effect of Autocorrelation Coefficient 𝜙 243

9.4 Summary and Concluding Remarks 245

10 Rank-Based Shrinkage Estimation 251

10.1 Introduction 251

10.2 LinearModel and Rank Estimation 252

10.2.1 Penalty R-Estimators 256

10.2.2 PTREs and Stein-type R-Estimators 258

10.3 Asymptotic Distributional Bias and L2 Risk of the R-Estimators 259

10.3.1 HardThreshold Estimators (Subset Selection) 259

10.3.2 Rank-based LASSO 260

10.3.3 Multivariate Normal DecisionTheory and Oracles for Diagonal Linear Projection 261

10.4 Comparison of Estimators 262

10.4.1 Comparison of RE with Restricted RE 262

10.4.2 Comparison of RE with PTRE 263

10.4.3 Comparison of RE with SRE and PRSRE 263

10.4.4 Comparison of RE and Restricted RE with RRRE 265

10.4.5 Comparison of RRRE with PTRE, SRE, and PRSRE 266

10.4.6 Comparison of RLASSO with RE and Restricted RE 267

10.4.7 Comparison of RLASSO with PTRE, SRE, and PRSRE 267

10.4.8 Comparison of Modified RLASSO with RRRE 268

10.5 Summary and Concluding Remarks 268

11 High-Dimensional Ridge Regression 285

11.1 High-Dimensional RRE 286

11.2 High-Dimensional Stein-Type RRE 288

11.2.1 Numerical Results 291

11.2.1.1 Example: Riboflavin Data 291

11.2.1.2 Monte Carlo Simulation 291

11.3 Post Selection Shrinkage 293

11.3.1 Notation and Assumptions 296

11.3.2 Estimation Strategy 297

11.3.3 Asymptotic Distributional L2-Risks 299

11.4 Summary and Concluding Remarks 300

12 Applications: Neural Networks and Big Data 303

12.1 Introduction 304

12.2 A Simple Two-Layer Neural Network 307

12.2.1 Logistic Regression Revisited 307

12.2.2 Logistic Regression Loss Function with Penalty 310

12.2.3 Two-Layer Logistic Regression 311

12.3 Deep Neural Networks 313

12.4 Application: Image Recognition 315

12.4.1 Background 315

12.4.2 Binary Classification 316

12.4.3 Image Preparation 318

12.4.4 Experimental Results 320

12.5 Summary and Concluding Remarks 323

References 325

Index 333