在單指標條件分配模型之下，我們提出擬最小平方估計法來估計模型中的單指標係數。數值實驗顯示此估計式的表現比傳統的擬最大概似估計式以及半參數化最小平方估計式要好。此外，根據所考量之資料結構，提出交互驗證法作為帶寬選取標準，同時借助自助重取法提供估計式的變異估計及信賴區間之建立。利用定義的殘差統計量，我們進一步建立模型適當性之檢定方法。當模型中之單指標有零係數發生狀況，多階段的Adaptive LASSO 演算法有效偵測出此類群變數。在數值方面，廣泛的模擬與實際資料之驗證呈現所提出方法之可行性。

A more flexible single-index regression model is employed to characterize the conditional distribution. For this emiparametric model, a pseudo least integrated squares pproach is developed for the estimation of index oefficients. It is shown in the numerical studies that our estimator outperforms both the pseudo maximum likelihood and semiparametric least squares ones. In addition, we propose the generalized cross-validation criteria for bandwidth selection and the bootstrap implementation for the estimation of asymptotic variance and the construction of confidence intervals. With our defined residual process, a test rule is established to check the adequacy of the considered single-index conditional distribution model. To tackle with the problem of sparse variables, a multiple-stage adaptive Lasso algorithm is developed to identify significant variables and achieve the semiparametric efficiency bound. In this study, a class of simulation scenarios was conducted to assess the finite sample properties of the proposed estimators and inference procedures. Two empirical examples from the house-price study in Boston and the environmental study in New York are further used to illustrate the usefulness of our approaches.

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
1 Introduction 1
2 Estimation and Inference Procedures 5
2.1 Estimation and Bandwidth Selection . . . . . . . . . . . . . . . . . . 5
2.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Bootstrap Inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Model Test and Sparse Models 14
3.1 Model Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Adaptive Lasso Estimation and Oracle Properties . . . . . . . . . . . 15
3.3 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Multiple-Stage Adaptive Lasso Procedure . . . . . . . . . . . . . . . 17
4 Simulation Experiments 19
4.1 Simulation I - Comparison of the Estimators . . . . . . . . . . . . . . 19
4.2 Simulation II - Assessment of Inference Procedures . . . . . . . . . . 20
4.3 Simulation III - Performance of Adaptive Lasso Estimation . . . . . . 22
5 Empirical Examples 28
5.1 Application to a Study of House Price . . . . . . . . . . . . . . . . . 28
5.2 Application to a Study of Air Quality . . . . . . . . . . . . . . . . . . 29
6 Concluding Remarks and Further Extensions 32
Reference 34

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