本研究討論使用具有空間隨機效應的迴歸模型在模型錯誤設定(model misspecification)下估計迴歸係數的估計表現。所考慮的模型包含空間自迴歸模型(Spatial Autoregressive Model)、高斯過程迴歸模型(Gaussian Process Model with Linear Regression)，以及局部線性迴歸模型(Partial Linear Regression Model)。
在模型具有不同空間效應下，推導出變數的迴歸係數及估計誤差的一般式與估計係數變異數。此外，本研究根據 Chang (2020) 論文中 PCBA 資料做模擬設計，討論了不同模型在模型錯誤設定下的估計誤差表現。
在此研究中，發現在真實模型為空間效應較弱的空間自迴歸模型情況下，以高斯過程迴歸模型作為擬合模型估計迴歸係數時，表現相當穩健。相反地，若在真實模型為具有較強空間效應的高斯過程迴歸模型情況，以空間自迴歸模型作為擬合模型估計迴歸係數時，則會具有顯著的估計誤差。

This thesis concerns the estimation properties on the regression coefficient estimates in regression models involving spatial random effects under model misspecification.
The models considered include spatial autoregressive models (SAR), regression models with Gaussian spatial processes, partial linear regressions.
Under different specifications for spatial effects, the estimation bias and variance for the regression coefficients on the covariates are derived theoretically. Moreover, the empirical performance on the estimation mean squared errors is investigated in a simulation study designed according to the PCBA data studied in Chang (2020).
This work found that in a SAR model with weak spatial influences,
this work found that the regression coefficient estimates are fairly robust against the misspecified GP modeling.
On the contrary, in a regression GP model with a wide range of spatial dependence,
the regression coefficient estimates have non-negligible efficiency loss when the fitted model is misspecified as a SAR model.

摘要 i
Abstract ii
List of Figures vi
List of Tables vii
1 Introduction 1
1.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review and Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . 2
2 Introduction to PCBA Data 4
2.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Definition of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Various Models for PCBA Data 6
3.1 Spatial Autoregressive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Gaussian Process Model with Linear Regression . . . . . . . . . . . . . . . . . . 7
3.3 Partial Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Estimation Bias and Variance 10
5 Simulation 13
5.1 Data Generation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Methods To Be Compared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.1 Performance Measures on Parameter Estimation . . . . . . . . . . . . . . 15
5.2.2 Performance Measures on Prediction . . . . . . . . . . . . . . . . . . . . 19
6 Application 22
7 Conclusion 24
References 25
Appendix : Results of Simulation Experiment with p = 6 26

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