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by B. M. Golam Kibria, Mohammad Arashi, A. K. Md. Ehsanes Saleh
Theory of Ridge Regression Estimation with Applications
Cover
Dedication
List of Figures
List of Tables
Preface
Abbreviations and Acronyms
List of Symbols
1 Introduction to Ridge Regression
1.1 Introduction
1.2 Ridge Regression Estimator: Ridge Notion
1.3 LSE vs. RRE
1.4 Estimation of Ridge Parameter
1.5 Preliminary Test and Stein‐Type Ridge Estimators
1.6 High‐Dimensional Setting
1.7 Notes and References
1.8 Organization of the Book
2 Location and Simple Linear Models
2.1 Introduction
2.2 Location Model
2.3 Simple Linear Model
2.4 Summary and Concluding Remarks
3 ANOVA Model
3.1 Introduction
3.2 Model, Estimation, and Tests
3.3 Bias and Weighted Risks of Estimators
3.4 Comparison of Estimators
3.5 Application
3.6 Efficiency in Terms of Unweighted Risk
3.7 Summary and Concluding Remarks
3.A Appendix
Problems
4 Seemingly Unrelated Simple Linear Models
4.1 Model, Estimation, and Test of Hypothesis
4.2 Bias and MSE Expressions of the Estimators
4.3 Comparison of Estimators
4.4 Efficiency in Terms of Unweighted Risk
4.5 Summary and Concluding Remarks
5 Multiple Linear Regression Models
5.1 Introduction
5.2 Linear Model and the Estimators
5.3 Bias and Weighted Risks of Estimators
5.4 Comparison of Estimators
5.5 Efficiency in Terms of Unweighted Risk
5.6 Summary and Concluding Remarks
6 Ridge Regression in Theory and Applications
6.1 Multiple Linear Model Specification
6.2 Ridge Regression Estimators (RREs)
6.3 Bias, MSE, and Risk of Ridge Regression Estimator
6.4 Determination of the Tuning Parameters
6.5 Ridge Trace
6.6 Degrees of Freedom of RRE
6.7 Generalized Ridge Regression Estimators
6.8 LASSO and Adaptive Ridge Regression Estimators
6.9 Optimization Algorithm
6.10 Estimation of Regression Parameters for Low‐Dimensional Models
6.11 Summary and Concluding Remarks
7 Partially Linear Regression Models
7.1 Introduction
7.2 Partial Linear Model and Estimation
7.3 Ridge Estimators of Regression Parameter
7.4 Biases and Risks of Shrinkage Estimators
7.5 Numerical Analysis
7.6 High‐Dimensional PLM
7.7 Summary and Concluding Remarks
8 Logistic Regression Model
8.1 Introduction
8.2 Asymptotic Distributional Risk Efficiency Expressions of the Estimators
8.3 Summary and Concluding Remarks
9 Regression Models with Autoregressive Errors
9.1 Introduction
9.2 Asymptotic Distributional ‐risk Efficiency Comparison
9.3 Example: Sea Level Rise at Key West, Florida
9.4 Summary and Concluding Remarks
10 Rank‐Based Shrinkage Estimation
10.1 Introduction
10.2 Linear Model and Rank Estimation
10.3 Asymptotic Distributional Bias and Risk of the R‐Estimators
10.4 Comparison of Estimators
10.5 Summary and Concluding Remarks
11 High‐Dimensional Ridge Regression
11.1 High‐Dimensional RRE
11.2 High‐Dimensional Stein‐Type RRE
11.3 Post Selection Shrinkage
11.4 Summary and Concluding Remarks
12 Applications: Neural Networks and Big Data
12.1 Introduction
12.2 A Simple Two‐Layer Neural Network
12.3 Deep Neural Networks
12.4 Application: Image Recognition
12.5 Summary and Concluding Remarks
References
Index
End User License Agreement
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Abbreviations and Acronyms
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1 Introduction to Ridge Regression
List of Symbols
matrices
and vectors
transpose of a vector or a matrix
expectation of a random variable
variance of a random variable
diagonal matrix
trace of a matrix
real numbers
response vector
design matrix
vector of unknown regression parameters
least squares estimator
restricted least squares estimator
preliminary test least squares estimator
James–Stein‐type least squares estimator
positive‐rule James–Stein‐type least squares estimator
error term
ridge regression estimator
restricted ridge regression estimator
preliminary test ridge regression estimator
James–Stein‐type ridge regression estimator
positive‐rule James–Stein‐type ridge regression estimator
generalized ridge regression estimator
shrinkage estimator of the location parameter
LASSO estimator of the location parameter
modified LASSO estimator of the location parameter
generalized least squares estimator
restricted generalized least squares estimator
preliminary test generalized least squares estimator
James–Stein‐type generalized least squares estimator
positive‐rule James–Stein‐type generalized least squares estimator
‐estimator
restricted
‐estimator
preliminary test
‐estimator
James–Stein‐type
‐estimator
positive‐rule James–Stein‐type
‐estimator
‐estimator
restricted
‐estimator
preliminary test
‐estimator
James–Stein‐type
‐estimator
positive‐rule James–Stein‐type
‐estimator
bias expression of an estimator
‐risk function of an estimator
c.d.f. of a standard normal distribution
c.d.f of a chi‐square distribution with
d.f. and noncentrality parameter
c.d.f. of a noncentral
distribution with
d.f. and noncentrality parameter
indicator function
identity matrix of order
normal distribution with mean
and variance
‐variate normal distribution with mean
and covariance
noncentrality parameter
covariance matrix of an estimator
sign function
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