Partially Linear Models

Editor: Wolfgang Hardle, Hua Liang, Jiti Gao.

preface

In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an up-to-date presentation of the state of the art of partially linear regression techniques. The emphasis of this monograph is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically ecient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models.
We hope that this monograph will serve as a useful reference for theoretical and applied statisticians and to graduate students and others who are interested in the area of partially linear regression. While advanced mathematical ideas have been valuable in some of the theoretical development, the methodological power of partially linear regression can be demonstrated and discussed without advanced mathematics.
This monograph can be divided into three parts: part one{Chapter 1 through Chapter 4; part two{Chapter 5; and part three{Chapter 6. In the first part, we discuss various estimators for partially linear regression models, establish theoretical results for the estimators, propose estimation procedures, and implement the proposed estimation procedures through real and simulated examples.
The second part is of more theoretical interest. In this part, we construct several adaptive and ecient estimates for the parametric component. We show that the LS estimator of the parametric component can be modi ed to have both Bahadur asymptotic eciency and second order asymptotic eciency.
In the third part, we consider partially linear time series models. First, we propose a test procedure to determine whether a partially linear model can be used to t a given set of data. Asymptotic test criteria and power investigations are presented. Second, we propose a Cross-Validation (CV) based criterion to select the optimum linear subset from a partially linear regression and establish a CV selection criterion for the bandwidth involved in the nonparametrickernel estimation. The CV selection criterion can be applied to the case where
the observations tted by the partially linear model (1.1.1) are independent and identically distributed (i.i.d.). Due to this reason, we have not provided a separate chapter to discuss the selection problem for the i.i.d. case. Third, we provide recent developments in nonparametric and semiparametric time series regression.

1 INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1
1.1 Background, History and Practical Examples . . . . . . . . . . . . 1
1.2 The Least Squares Estimators . . . . . . . . . . . . . . . . . . . . 12
1.3 Assumptions and Remarks . . . . . . . . . . . . . . . . . . . . . . 14
1.4 The Scope of the Monograph . . . . . . . . . . . . . . . . . . . . . 16
1.5 The Structure of the Monograph . . . . . . . . . . . . . . . . . . . 17
2 ESTIMATION OF THE PARAMETRIC COMPONENT : : :: :: :: :: :: :: :: :: :: : 19
2.1 Estimation with Heteroscedastic Errors . . . . . . . . . . . . . . . 19
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Estimation of the Non-constant Variance Functions . . . . . . . .. . 22
2.1.3 Selection of Smoothing Parameters . . . . . . . . .  . . . . . . . . 26
2.1.4 Simulation Comparisons . . . . . . . . . . . . . . .  . . . .. . . . 27
2.1.5 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Estimation with Censored Data . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 Synthetic Data and Statement of the Main Results . .  . . . .. . 33
2.2.3 Estimation of the Asymptotic Variance . . . . . . . .  . . . .. . . 37
2.2.4 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . 37
2.2.5 Technical Details . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Bootstrap Approximations . . . . . . . . . . . . . . . . . . . . . . 41
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2 Bootstrap Approximations . . . . . . . . . . . . . . . . . . . . . . 42
2.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 43
3 ESTIMATION OF THE NONPARAMETRIC COMPONENT ::  ::  ::  ::  ::  ::  ::  45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Consistency Results . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Simulated and Real Examples . . . . . . . . . . . . . . . . . . . . 50
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 ESTIMATION WITH MEASUREMENT ERRORS : : : : :  ::  ::  ::  ::  ::  :: : 55
4.1 Linear Variables with Measurement Errors . . . . . . . . . . . . . 55
4.1.1 Introduction and Motivation . . . . . . . . . . . . . . . . . 55
4.1.2 Asymptotic Normality for the Parameters . . . . . . . . . 56
4.1.3 Asymptotic Results for the Nonparametric Part . . . . . . 58
4.1.4 Estimation of Error Variance . . . . . . . . . . . . . . . . 58
4.1.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . 59
4.1.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.7 Technical Details . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Nonlinear Variables with Measurement Errors . . . . . . . . . . . 65
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.2 Construction of Estimators . . . . . . . . . . . . . . . . . . 66
4.2.3 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . 67
4.2.4 Simulation Investigations . . . . . . . . . . . . . . . . . . . 68
4.2.5 Technical Details . . . . . . . . . . . . . . . . . . . . . . . 70
5 SOME RELATED THEORETIC TOPICS : : : : : : : : : ::  ::  ::  : : : : 77
5.1 The Laws of the Iterated Logarithm . . . . . . . . . . . . . . . . . 77
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.2 Preliminary Processes . . . . . . . . . . . . . . . . . . . . 78
5.1.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 The Berry-Esseen Bounds . . . . . . . . . . . . . . . . . . . . . . 82
5.2.1 Introduction and Results . . . . . . . . . . . . . . . . . . . 82
5.2.2 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.3 Technical Details . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Asymptotically Ecient Estimation . . . . . . . . . . . . . . . . . 94
5.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.2 Construction of Asymptotically Ecient Estimators . . . . 94
5.3.3 Four Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Bahadur Asymptotic Eciency . . . . . . . . . . . . . . . . . . . 104
5.4.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.2 Tail Probability . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4.3 Technical Details . . . . . . . . . . . . . . . . . . . . . . . 106
5.5 Second Order Asymptotic Eciency . . . . . . . . . . . . . . . . . 111
5.5.1 Asymptotic Eciency . . . . . . . . . . . . . . . . . . . . 111
5.5.2 Asymptotic Distribution Bounds . . . . . . . . . . . . . . 113
5.5.3 Construction of 2nd Order Asymptotic Ecient Estimator 117
5.6 Estimation of the Error Distribution . . . . . . . . . . . . . . . . 119
5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.6.2 Consistency Results . . . . . . . . . . . . . . . . . . . . . . 120
5.6.3 Convergence Rates . . . . . . . . . . . . . . . . . . . . . . 124
5.6.4 Asymptotic Normality and LIL . . . . . . . . . . . . . . . 125
6 PARTIALLY LINEAR TIME SERIES MODELS : : : : : : : ::  ::  ::  : 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Adaptive Parametric and Nonparametric Tests . . . . . . . . . . . 127
6.2.1 Asymptotic Distributions of Test Statistics . . . . . . . . . 127
6.2.2 Power Investigations of the Test Statistics . . . . . . . . . 131
6.3 Optimum Linear Subset Selection . . . . . . . . . . . . . . . . . . 136
6.3.1 A Consistent CV Criterion . . . . . . . . . . . . . . . . . . 136
6.3.2 Simulated and Real Examples . . . . . . . . . . . . . . . . 139
6.4 Optimum Bandwidth Selection . . . . . . . . . . . . . . . . . . . 144
6.4.1 Asymptotic Theory . . . . . . . . . . . . . . . . . . . . . . 144
6.4.2 Computational Aspects . . . . . . . . . . . . . . . . . . . . 150
6.5 Other Related Developments . . . . . . . . . . . . . . . . . . . . . 156
6.6 The Assumptions and the Proofs of Theorems . . . . . . . . . . . 157
6.6.1 Mathematical Assumptions . . . . . . . . . . . . . . . . . 157
6.6.2 Technical Details . . . . . . . . . . . . . . . . . . . . . . . 160
APPENDIX: BASIC LEMMAS : : : : : : : : : : : : : : : : : : : : : 183
REFERENCES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 187

好书不解释，对想了解半参数部分线性模型的同学会有一些帮助的！