Theory of Ridge Regression Estimation with Applications

Name: Theory of Ridge Regression Estimation with Applications
Brand: Wiley-ISTE
Price: 111.99 EUR
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A. K. Md. Ehsanes Saleh Mohammad Arashi B. M. Golam Kibria(Autor*in)

Wiley-ISTE (Verlag)

1. Auflage

Erschienen am 8. Januar 2019

384 Seiten

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Inhalt

Cover
Title Page
Copyright
Contents
List of Figures
List of Tables
Preface
Abbreviations and Acronyms
List of Symbols
Chapter 1 Introduction to Ridge Regression
1.1 Introduction
1.1.1 Multicollinearity Problem
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
Chapter 2 Location and Simple Linear Models
2.1 Introduction
2.2 Location Model
2.2.1 Location Model: Estimation
2.2.2 Shrinkage Estimation of Location
2.2.3 Ridge Regression-Type Estimation of Location Parameter
2.2.4 LASSO for Location Parameter
2.2.5 Bias and MSE Expression for the LASSO of Location Parameter
2.2.6 Preliminary Test Estimator, Bias, and MSE
2.2.7 Stein-Type Estimation of Location Parameter
2.2.8 Comparison of LSE, PTE, Ridge, SE, and LASSO
2.3 Simple Linear Model
2.3.1 Estimation of the Intercept and Slope Parameters
2.3.2 Test for Slope Parameter
2.3.3 PTE of the Intercept and Slope Parameters
2.3.4 Comparison of Bias and MSE Functions
2.3.5 Alternative PTE
2.3.6 Optimum Level of Significance of Preliminary Test
2.3.7 Ridge-Type Estimation of Intercept and Slope
2.3.7.1 Bias and MSE Expressions
2.3.8 LASSO Estimation of Intercept and Slope
2.4 Summary and Concluding Remarks
Chapter 3 ANOVA Model
3.1 Introduction
3.2 Model, Estimation, and Tests
3.2.1 Estimation of Treatment Effects
3.2.2 Test of Significance
3.2.3 Penalty Estimators
3.2.4 Preliminary Test and Stein-Type Estimators
3.3 Bias and Weighted L2 Risks of Estimators
3.3.1 Hard Threshold Estimator (Subset Selection Rule)
3.3.2 LASSO Estimator
3.3.3 Ridge Regression Estimator
3.4 Comparison of Estimators
3.4.1 Comparison of LSE with RLSE
3.4.2 Comparison of LSE with PTE
3.4.3 Comparison of LSE with SE and PRSE
3.4.4 Comparison of LSE and RLSE with RRE
3.4.5 Comparison of RRE with PTE, SE, and PRSE
3.4.5.1 Comparison Between ^ RR n (kopt) and ^ PT n ( )
3.4.5.2 Comparison Between ^ RRn (kopt) and ^ Sn
3.4.5.3 Comparison of ^ RRn (kopt) with ^ S+n
3.4.6 Comparison of LASSO with LSE and RLSE
3.4.7 Comparison of LASSO with PTE, SE, and PRSE
3.4.8 Comparison of LASSO with RRE
3.5 Application
3.6 Efficiency in Terms of Unweighted L2 Risk
3.7 Summary and Concluding Remarks
3A. Appendix
Chapter 4 Seemingly Unrelated Simple Linear Models
4.1 Model, Estimation, and Test of Hypothesis
4.1.1 LSE of ? and ß
4.1.2 Penalty Estimation of ß and ?
4.1.3 PTE and Stein-Type Estimators of ß and ?
4.2 Bias and MSE Expressions of the Estimators
4.3 Comparison of Estimators
4.3.1 Comparison of LSE with RLSE
4.3.2 Comparison of LSE with PTE
4.3.3 Comparison of LSE with SE and PRSE
4.3.4 Comparison of LSE and RLSE with RRE
4.3.5 Comparison of RRE with PTE, SE, and PRSE
4.3.5.1 Comparison Between ^ RRn (kopt) and ^ PTn
4.3.5.2 Comparison Between ^ RRn (kopt) and ^ Sn
4.3.5.3 Comparison of ^ RRn (kopt) with ^ S+n
4.3.6 Comparison of LASSO with RRE
4.3.7 Comparison of LASSO with LSE and RLSE
4.3.8 Comparison of LASSO with PTE, SE, and PRSE
4.4 Efficiency in Terms of Unweighted L2 Risk
4.4.1 Efficiency for ß
4.4.2 Efficiency for ?
4.5 Summary and Concluding Remarks
Chapter 5 Multiple Linear Regression Models
5.1 Introduction
5.2 Linear Model and the Estimators
5.2.1 Penalty Estimators
5.2.2 Shrinkage Estimators
5.3 Bias and Weighted L2 Risks of Estimators
5.3.1 Hard Threshold Estimator
5.3.2 Modified LASSO
5.3.3 Multivariate Normal Decision Theory and Oracles for Diagonal Linear Projection
5.3.4 Ridge Regression Estimator
5.3.5 Shrinkage Estimators
5.4 Comparison of Estimators
5.4.1 Comparison of LSE with RLSE
5.4.2 Comparison of LSE with PTE
5.4.3 Comparison of LSE with SE and PRSE
5.4.4 Comparison of LSE and RLSE with RRE
5.4.5 Comparison of RRE with PTE, SE, and PRSE
5.4.5.1 Comparison Between ^ RRn (kopt) and ^ PTn ( )
5.4.5.2 Comparison Between ^ RRn (kopt) and ^ Sn
5.4.5.3 Comparison of ^ RRn (kopt) with ^ S+n
5.4.6 Comparison of MLASSO with LSE and RLSE
5.4.7 Comparison of MLASSO with PTE, SE, and PRSE
5.4.8 Comparison of MLASSO with RRE
5.5 Efficiency in Terms of Unweighted L2 Risk
5.6 Summary and Concluding Remarks
Chapter 6 Ridge Regression in Theory and Applications
6.1 Multiple Linear Model Specification
6.1.1 Estimation of Regression Parameters
6.1.2 Test of Hypothesis for the Coefficients Vector
6.2 Ridge Regression Estimators (RREs)
6.3 Bias, MSE, and L2 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
6.9.1 Prostate Cancer Data
6.10 Estimation of Regression Parameters for Low-Dimensional Models
6.10.1 BLUE and Ridge Regression Estimators
6.10.2 Bias and L2-risk Expressions of Estimators
6.10.3 Comparison of the Estimators
6.10.4 Asymptotic Results of RRE
6.11 Summary and Concluding Remarks
Chapter 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 L2 Risks of Shrinkage Estimators
7.5 Numerical Analysis
7.5.1 Example: Housing Prices Data
7.6 High-Dimensional PLM
7.6.1 Example: Riboflavin Data
7.7 Summary and Concluding Remarks
Chapter 8 Logistic Regression Model
8.1 Introduction
8.1.1 Penalty Estimators
8.1.2 Shrinkage Estimators
8.1.3 Results on MLASSO
8.1.4 Results on PTE and Stein-Type Estimators
8.1.5 Results on Penalty Estimators
8.2 Asymptotic Distributional L2 Risk Efficiency Expressions of the Estimators
8.2.1 MLASSO vs. MLE
8.2.2 MLASSO vs. RMLE
8.2.3 Comparison of MLASSO vs. PTE
8.2.4 PT and MLE
8.2.5 Comparison of MLASSO vs. SE
8.2.6 Comparison of MLASSO vs. PRSE
8.2.7 RRE vs. MLE
8.2.7.1 RRE vs. RMLE
8.2.8 Comparison of RRE vs. PTE
8.2.9 Comparison of RRE vs. SE
8.2.10 Comparison of RRE vs. PRSE
8.2.11 PTE vs. SE and PRSE
8.2.12 Numerical Comparison Among the Estimators
8.3 Summary and Concluding Remarks
Chapter 9 Regression Models with Autoregressive Errors
9.1 Introduction
9.1.1 Penalty Estimators
9.1.2 Shrinkage Estimators
9.1.2.1 Preliminary Test Estimator
9.1.2.2 Stein-Type and Positive-Rule Stein-Type Estimators
9.1.3 Results on Penalty Estimators
9.1.4 Results on PTE and Stein-Type Estimators
9.1.5 Results on Penalty Estimators
9.2 Asymptotic Distributional L2-risk Efficiency Comparison
9.2.1 Comparison of GLSE with RGLSE
9.2.2 Comparison of GLSE with PTE
9.2.3 Comparison of LSE with SE and PRSE
9.2.4 Comparison of LSE and RLSE with RRE
9.2.5 Comparison of RRE with PTE, SE and PRSE
9.2.5.1 Comparison Between ^ GRRn (kopt) and ^ G(PT)n
9.2.5.2 Comparison Between ^ GRRn (kopt) and ^ G(S)n
9.2.5.3 Comparison of ^ GRRn (kopt) with ^ G(S+)n
9.2.6 Comparison of MLASSO with GLSE and RGLSE
9.2.7 Comparison of MLASSO with PTE, SE, and PRSE
9.2.8 Comparison of MLASSO with RRE
9.3 Example: Sea Level Rise at Key West, Florida
9.3.1 Estimation of the Model Parameters
9.3.1.1 Testing for Multicollinearity
9.3.1.2 Testing for Autoregressive Process
9.3.1.3 Estimation of Ridge Parameter k
9.3.2 Relative Efficiency
9.3.2.1 Relative Efficiency (REff)
9.3.2.2 Effect of Autocorrelation Coefficient ?
9.4 Summary and Concluding Remarks
Chapter 10 Rank-Based Shrinkage Estimation
10.1 Introduction
10.2 Linear Model and Rank Estimation
10.2.1 Penalty R-Estimators
10.2.2 PTREs and Stein-type R-Estimators
10.3 Asymptotic Distributional Bias and L2 Risk of the R-Estimators
10.3.1 Hard Threshold Estimators (Subset Selection)
10.3.2 Rank-based LASSO
10.3.3 Multivariate Normal Decision Theory and Oracles for Diagonal Linear Projection
10.4 Comparison of Estimators
10.4.1 Comparison of RE with Restricted RE
10.4.2 Comparison of RE with PTRE
10.4.3 Comparison of RE with SRE and PRSRE
10.4.4 Comparison of RE and Restricted RE with RRRE
10.4.5 Comparison of RRRE with PTRE, SRE, and PRSRE
10.4.6 Comparison of RLASSO with RE and Restricted RE
10.4.7 Comparison of RLASSO with PTRE, SRE, and PRSRE
10.4.8 Comparison of Modified RLASSO with RRRE
10.5 Summary and Concluding Remarks
Chapter 11 High-Dimensional Ridge Regression
11.1 High-Dimensional RRE
11.2 High-Dimensional Stein-Type RRE
11.2.1 Numerical Results
11.2.1.1 Example: Riboflavin Data
11.2.1.2 Monte Carlo Simulation
11.3 Post Selection Shrinkage
11.3.1 Notation and Assumptions
11.3.2 Estimation Strategy
11.3.3 Asymptotic Distributional L2-Risks
11.4 Summary and Concluding Remarks
Chapter 12 Applications: Neural Networks and Big Data
12.1 Introduction
12.2 A Simple Two-Layer Neural Network
12.2.1 Logistic Regression Revisited
12.2.2 Logistic Regression Loss Function with Penalty
12.2.3 Two-Layer Logistic Regression
12.3 Deep Neural Networks
12.4 Application: Image Recognition
12.4.1 Background
12.4.2 Binary Classification
12.4.3 Image Preparation
12.4.4 Experimental Results
12.5 Summary and Concluding Remarks
References
Index
EULA

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