非参数和半参数模型中的经验似然

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

5.1 Introduction

5.2 Naive EL and maximum EL estimation

5.2.1 Wilks phenomenon of naive 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 Naive 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 Reflned 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 Adiusted 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 coe~cients

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

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 readers 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.