二元分類一直是統計分析或監督式學習中值得被討論的內容。在建立二元結果與變數的模型選擇上,logistic 與 probit 的模型是較常被使用的。然而,在資料維度遽增以及不可忽視的測量誤差存在測量結果、變數當中,過去的傳統方法已不適用,這為我們在資料分析上帶來了重大的挑戰。為了解決上述的問題,我們提出有效的推論方法處理測量誤差並同時進行變數選取。具體來說,我們首先考慮 logistic 或 probit 的模型,將經過校正的應變數與自變數放入我們的估計函數中。接著,我們透過 boosting 方法去做變數選取並計算參數的估計值。在數值研究當中,我們所提出的方法能夠準確地保留重要變數且能精準地計算出估計參數。此外,經過誤差校正的結果在整體的分析表現上是顯著優於沒有校正的結果。

Binary classification has been an attractive topic in statistical analysis or supervised learning. To model a binary response and predictors, logistic regression models or probit models are perhaps commonly used approaches. However, because of the rapid growth of the dimension of the data as well as the non ignorability of measurement error in responses and/or predictors, data analysis becomes challenging and conventional methods are invalid. To address those concerns, we propose a valid inferential method to deal with measurement error and handle variable selection simultaneously. Specifically, we primarily consider logistic regression models or probit models, and propose corrected estimating functions by incorporating error-eliminated responses and predictors. After that, we develop the boosting procedure with corrected estimating functions accommodated to do variable selection and estimation.Through numerical studies, we find that the proposed method accurately retains informative predictors as well as gives precise estimators, and its performance is generally better than that without measurement error correction.

Chapter 1 Introduction 1
Chapter 2 Notation and Models 3
2.1 Data Structure 3
2.2 Measurement Error Models 4
Chapter 3 Methodology 6
3.1 Correction of Measurement Error Effects 6
3.2 Variable Selection via Boosting 8
3.3 Extension: Collinearity 11
Chapter 4 Estimation with Validation Data 11
4.1 BOOME via External Validation 12
4.2 BOOME via Internal Validation 13
Chapter 5 Python Package: BOOME 14
5.1 ME_Generate 14
5.2 LR_Boost 15
5.3 PM_Boost 16
Chapter 6 Numerical Studies 16
6.1 Simulation Setup 16
6.2 Simulation Results 17
6.3 Simulation Results based on Validation Data 19
6.4 Analysis of Bankruptcy Data 19
Chapter 7 Summary 21
References 23
Appendix 26
A.1 Proof of Theorem 3.1 26
A.2 Proof of Theorem 3.2 28
A.3 Proof of Theorem 3.3 29

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