Improving Efficiency by Shrinkage: The James-Stein and Ridge Regression Estimators (Statistics: A Series of Textbooks and Monographs)

Preface......Page 14
Table of Contents......Page 18
1.0 Motivation for Writing This Book......Page 23
1.1 Purpose of This Book......Page 26
1.2 Introduction to the James-Stein Estimator (JS)......Page 27
1.3 Introduction to the Ridge Regression Estimator......Page 29
1.4 Historical Survey......Page 37
1.5 Empirical Bayes Methodology......Page 83
1.6 A Minimum Mean Square Error Estimator, Its Approximation as an Empirical Bayes Estimator......Page 88
1.7 The Structure of This Book......Page 90
2.0 Estimation of the Mean......Page 93
2.1 The Multivariate Normal Distribution......Page 95
2.2 Maximum Likelihood Estimation......Page 98
2.3 The Decision Theory Framework......Page 103
2.4 Why the MLE May Not Be the Best......Page 106
2.5 Another Case Against the MLE......Page 113
2.7 The JS as an Empirical Bayes Estimator (EBE)......Page 121
2.8. How the JS is a Special Case of the Operational Ridge Regression Estimator......Page 126
2.9 Introduction to the Positive Part of the JS......Page 129
2.10 Summary......Page 132
3.0 The Need for Alternatives to the Least Square Estimators......Page 133
3.1 Derivation of the Ridge Estimator......Page 134
3.2 The Efficiency of the Ridge Regression Estimator......Page 139
3.3 Estimating the Ridge Perturbation Factor k- The Ridge Trace......Page 155
3.4 Estimating the Ridge Perturbation Factor k- Objective Methods......Page 163
3.5 Evaluation of the Efficiency of Ridge Estimators by Computer Simulation......Page 180
3.6 Summary......Page 188
4.0 Introduction......Page 189
4.1 Review of Basic Linear Model Concepts......Page 191
4.2 The James-Stein Estimator from a Bayesian Point of View......Page 195
4.3 The Non-Bayesian Formulation of the JS and the JSL......Page 208
4.4 The Average Mean Square Error (MSE)......Page 212
4.5 The Conditional MSE......Page 217
4.6 The MSE of the JS for Different Quadratic Loss Functions......Page 224
4.7 The Efficiency of the JS as Compared with Optimum Estimators......Page 228
4.9 Appendix......Page 245
5.0 Introduction......Page 249
5.1 Some More Review of Linear Models......Page 251
5.2 The Generalized Ridge Regression Estimator......Page 262
5.3 The Mixed Estimators......Page 264
5.4 The Linear Minimax Estimator......Page 270
5.5 The Linear Bayes Estimator......Page 272
5.6 Comparing the Efficiency of the Ridge and the Least Square Estimator from the Frequentist Point of View......Page 288
5.7 Comparing the MSE of Ridge-Type Estimators from the Bayesian Point of View......Page 306
5.8 The Jack-knifed Ridge Estimator......Page 320
5.9 Summary......Page 327
6.0 Introduction......Page 329
6.1 The Positive Parts of the James-Stein Estimator (PP0 - PP4)......Page 331
6.2 The Average Mean Square Error......Page 334
6.3 The Conditional Mean Square Error......Page 359
6.4 The Estimators of Shao and Strawderman......Page 386
6.5 Summary......Page 392
7.0 Introduction......Page 393
7.1 TheEBEofC.R.Rao......Page 395
7.2 The EBE of Wind......Page 409
7.3 The Estimator of Dempster......Page 425
7.4 A Generalization of the EBE......Page 435
7.5 Comparing the Efficiency of the Estimators......Page 440
7.6 The Positive Parts......Page 454
7.7 Summary......Page 461
8.0 Introduction......Page 463
8.1 The Mean of a Multivariate Normal Distribution......Page 464
8.2 The Single Linear Model......Page 470
8.3 The Case of r Linear Models......Page 480
8.4 The Limited Translation Estimators......Page 502
8.5 Summary......Page 512
9.0 Introduction......Page 513
9.1 The Multivariate Linear Model......Page 515
9.2 The BE......Page 518
9.3 The EBE of Wind for One Linear Model......Page 522
9.4 The EBE for r Linear Models......Page 534
9.5 The Estimator of Dempster (An Example of an Approximate MSE)......Page 547
9.6 Summary......Page 551
10.0 Introduction......Page 553
10.1 The Seemingly Unrelated Regression Model (SURE)......Page 555
10.2 Some Aspects of the Simultaneous Estimation Problem......Page 564
10.3 The Kalman Filter......Page 578
10.4 Summary......Page 600
11.0 Introduction......Page 601
11.1 Overview of the Chapters......Page 602
11.2 Conclusion......Page 612
References......Page 613
Author Index......Page 641
Subject Index......Page 647