本論文主要是在探討當二維資料 (X,Y) 中的解釋變數 X 具有測量誤差 (measurement errors) 時，該怎麼對資料進行處理與估計。當我們觀測不到實際的數據，只能觀測到有測量誤差的數據時，應該如何對平常使用的估計方法做一些修正。在本論文中，我們使用的估計方法為 local linear regression，並且結合了 asymptotic projection matrix，對矩陣當中的核函數進行一些修正。本文主要針對我們提出的估計方法推導出理論性質，我們得出了 bias 的 order 為 h^4，比平常使用的估計方法 bias 的 order h^2 還要小。在模擬中也可以發現到這個現象，當資料本身的測量誤差沒有很大的時候，我們提出的估計方法，MSE大致上會比其他方法小。在實際資料中，也能發現到我們的估計方法表現不錯，而且估出來的迴歸曲線更貼合資料的趨勢。

The main focus of this thesis is to discuss estimation when the explanatory variable X of the (X,Y) data has measurement errors. When the actual values of X are un-observable and only data with measurement errors are available, it is important to adjust statistical estimation methods. We adopt local linear regression for estimation. We incorporate the asymptotic projection matrix and make some adjustments to the kernel function. This thesis primarily derives the theoretical properties of our proposed estimation method.
We find that our method has a bias of order h^4, which is smaller than the usual order $h^2$. This phenomenon is also observed in the simulations; in some scenarios, the Mean Squared Errors (MSE) of our proposed estimation method are generally smaller than those of existing methods. Furthermore, in the real data analysis, our estimation method also performs well, and the estimated regression curve fits the data more closely.

1 Introduction 4
2 Background 5
2.1 Local linear regression 5
2.2 Local linear estimator for errors-in-variables problem 6
2.3 Asymptotic projection matrix H* 8
2.4 H*Y estimator for the errors-in-variables problem 9
2.5 Linear regression with errors in the variables 10
3 Method 11
3.1 The conditional bias of Y* 13
3.2 The conditional variance of Y* 15
3.3 Lemmas 17
3.4 Kernel normalization 23
4 Simulation study 25
4.1 Simulation 1 26
4.2 Simulation 2 30
4.3 Simulation 3 34
5 Real data analysis 39
5.1 Data 2000 39
5.2 Data 2021 43
6 Concluding remarks 46

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