内容简介
This book is composed of ten chapters. The first chapter contains the preliminary knowledge about empirical likelihood and other relevant nonparametric methods. Chapters 2 and 3 analyze the section-data using the single-index model and the partially linear single-index model. Chapters 4 through 6 investigate the longitudinal data using the partially linear model, the varying coefficient model and a nonparametric regression model. Chapter 7 discusses nonlinear errors-in-covariables models with validation data. Chapters 8 through 10 investigate missing data under the framework of the linear model, a nonparametric regression model and the partially linear model. Every chapter, except for Chapter 1, of this book is self-contained so that the reader could focus on any chapter without much effect on the understanding of the others, and hence can read any chapters according to reader's own interest. The emphasis of this book is on methodologies rather than on theory, with a particular focus on applications of the empirical likelihood techniques to various semiparametric regression models. Key technical arguments are presented in the "proofs sections" at the end of each chapter. This gives interested researchers an idea of how the theoretical results are obtained. Also from the style of material organization, this book is more likely a lecture note, rather than a textbook. Most materials come from authors' research articles.
This book intends to provide a useful reference for researchers and to serve as a lecture note to postgraduate students. It is especially for the people working in the nonparametric and semiparametric statistics areas or applying the empirical likelihood method to other areas. 目录
Preface
Chapter 1 Preliminary knowledge
1.1 Empirical likelihood (EL)
1.1.1 Definition of EL
1.1.2 EL for mean
1.1.3 Estimating equations
1.1.4 Advantages of EL
1.1.5 Related literature
1.2 Bootstrap method
1.3 Smoothing methods
1.3.1 The Nadaraya-Watson estimator
1.3.2 The local polynomial smoother
1.4 Cross-validation
1.4.1 Least squares cross-validation
1.4.2 Generalized cross-validation
1.5 Data sets
1.5.1 Longitudinal data
1.5.2 Measurement error data
1.5.3 Missing data
1.6 Some notations
.Chapter 2 EL for single-index models
2.1 Introduction
2.2 Methods and results
2.2.1 Estimated EL
2.2.2 Two adjusted EL ratios
2.3 Simulation results
2.4 Proofs
2.4.1 Some lemmas
2.4.2 Proofs of theorems
Chapter 3 EL in a partially linear single-index model
3.1 Introduction
3.2 Methodology
3.2.1 The general case
3.2.2 Two special cases: single-index model and partially linear model
3.3 Simulation results
3.3.1 Preamble
3.3.2 Simulated examples
3.3.3 A real example
3.4 Proofs
3.4.1 A brief description of the proofs
3.4.2 Proofs of theorems
Chapter 4 EL semiparametric regression analysis
4.1 Introduction
4.2 Maximum EL estimator
4.2.1 Estimating the regression coefficients
4.2.2 Estimating the baseline function
4.3 Confidence regions for regression coefficients
4.3.1 Confidence regions based on normal approximation
4.3.2 EL confidence region
4.4 Confidence intervals for baseline function
4.4.1 Normal approximation-based confidence interval
4.4.2 Mean-corrected EL confidence interval
4.4.3 Residual-adjusted EL confidence interval
4.5 Numerical results
4.5.1 Bandwidth choice
4.5.2 Simulation studies
4.5.3 An application
4.6 Proofs
Chapter 5 EL for a varying coefficient model
5.I Introduction
5.2 ** EL and maximum EL estimation
5.2.1 Wilks' phenomenon of ** EL
5.2.2 Equivalence between MELE and WLSE
5.3 Two bias corrections
5.3.1 Mean-corrected EL
5.3.2 Residual-adjusted EL
5.4 Asymptotic confidence regions
5.4.1 The general cases
5.4.2 Partial profile EL for confidence intervals
5.4.3 Simultaneous confidence bands
5.4.4 Confidence regions based on the normal approximation
5.4.5 Bootstrap confidence intervals and bands
5.5 Numerical results
5.5.1 Bandwidth choice
5.5.2 Simulation studies
5.5.3 The application to AIDS data
5.6 Proofs of Theorems
Chapter 6 EL local polynomial regression analysis
6.1 Introduction
6.2 ** empirical likelihood
6.2.1 Prime method
6.2.2 Asymptotic properties
6.3 A bias correction method
6.4 Asymptotic confidence regions
6.4.1 Confidence regions based on EL
6.4.2 Pointwise confidence intervals based on partial EL
6.4.3 Confidence regions based on the normal approximation.
6.4.4 Simultaneous confidence band
6.5 Bandwidth selection
6.5.1 Pilot bandwidth selection
6.5.2 Refined bandwidth selection
6.5.3 Undersmoothing bandwidth selection
6.6 Numerical results
6.6.1 Simulation study
6.6.2 A real example
6.7 Concluding remarks
6.8 Proofs of Theorems
Chapter 7 EL in nonlinear EV models
7.1 Introduction
7.2 Estimated EL
7.3 Adjusted EL
7.4 Simulations and application
7.4.1 Simulations
7.4.2 A real data example
7.5 Conclusions
7.6 Proofs
Chapter 8 EL for the linear models
8.1 Introduction
8.2 EL for the regression coefficients
8.2.1 EL with complete-case data
8.2.2 Weighted EL
8.2.3 EL with the imputed values
8.2.4 Asymptotic properties
8.3 EL for the response mean
8.3.1 Weight-corrected EL
8.3.2 Normal approximation
8.4 Simulations
8.4.1 One dimensional case
8.4.2 Two dimensional case
8.4.3 A real example
8.5 Concluding remarks
8.6 Proofs
Chapter 9 EL for response mean
9.1 Introduction
9.2 Methods and results
9.2.1 Weight-corrected EL
9.2.2 Weight-corrected EL with auxiliary information
9.2.3 Normal approximation-based method
9.3 Simulations
9.4 Concluding remarks
9.5 Proofs
Chapter 10 EL for a semiparametric regression model
10.1 Introduction
10.2 EL for the regression coefficients
10.2.1 EL with complete-case data
10.2.2 EL with the imputed values
10.2.3 Partial profile empirical likelihood
10.3 EL for the baseline function
10.3.1 Estimated EL
10.3.2 Residual-adjusted EL
10.3.3 Simultaneous confidence band
10.4 EL for the response mean
10.4.1 Weight-corrected EL
10.4.2 Normal approximation
10.5 Simulations
10.5.1 One-dimensional case
10.5.2 Two-dimensional case
10.6 Application
10.7 Concluding remarks
10.8 Proofs
References
Index
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