迴歸分析(Regression analysis)是一種了解解釋變數與反應變數間之數量關係的統計分析方法(statistical method)。建構迴歸分析時，有時會碰到資料中的解釋變數無法準確測量，我們稱這種模式為測量誤差模式〈measurement error model〉。測量誤差模式中的迴歸分析，在解釋變數獨立於測量誤差跟迴歸誤差，且測量誤差跟迴歸誤差為二維常態(bivariate normal)，時參數是不可確認。因此在測量誤差模式中，最基本的問題是甚麼樣的條件會使得模式的參數可確認〈identifiable〉。在本文我們假設測量誤差為常態分佈有未知變異數且期望值為0，對簡單線性迴歸模型及二次迴歸模型探討其真實解釋變數參數可確認性的條件，另外提出俱一致性的參數估計量，並模擬研究其MSE的表現情況。

Regression analysis is a statistic method for understanding the relationship between independent variable and dependent variable. When establishing the Regression Analysis, sometimes people will meet the problem that the independent variable in the data base cannot be measure exactly, and that is what people called measurement error model. The reason to determine the regression analysis of measurement error model is to explain when the variable becomes independent in measurement error and regression error, and the measurement error and regression error are bivariate normal, the parameter is unidentifiable. Therefore, in the measurement error model, the basic problem is what conditions can make the parameters to be identifiable. In this thesis, we suppose that the measurement error is normal distribution with mean zero and unknown variable, and discuss the identifiable qualifications of parameters in simple linear regression model and in quadratic polynomial model respectively. We also discuss the estimators of parameters which are consistency, and simulate the performance of mean square error.

目錄
第一章  緒論…………………………………………………………………………1
第二章  解釋變數與反應變數的動差關係與模型的可辨認性……………………3
2.1簡單線性迴歸的參數的可辨認性…………………………………………3
2.1.1測量誤差為對稱分佈時參數的不可辨認性…………………………3
2.1.2測量誤差為常態分佈時參數的可辨認性……………………………4
2.2二次迴歸的參數的可辨認性………………………………………………6
2.2.1測量誤差為常態分佈時參數的可辨認性……………………………6
2.2.1.1當真實解釋變數為常態分佈……………………………………7
2.2.1.2當真實解釋變數為(-1,1)的均勻分佈…………………………8
2.2.1.3當真實解釋變數為指數分佈……………………………………9
2.2.2當真實解釋變數為對稱分佈………………………………………..10
2.2.3當截距項β0已知……………………………………………………..12
第三章  一致性估計量與電腦模擬結果…………………………………………..15
3.1簡單線性迴歸的參數模擬估計…………………………………………..15
3.1.1 Naive est. …………………………………………………………….15
3.1.2已知測量誤差變異數訊息時的參數估計…...………………………16
3.1.3使用高階動差估計…………………………………………………..17
3.2二次迴歸的參數模擬估計………………………………………………..19
3.2.1在真實解釋變數為對稱分佈且非為常態分佈時二次迴歸的參數估計……………………………………………………………………20
3.2.2在真實解釋變數為常態分佈且假設β0為已知的情況下二次迴歸的參數估計……………………………………………………..….….20
3.2.3 在β0為已知下對二次迴歸的參數估計…………………..………..21
第四章  結論………………………………………………………………………..22
參考文獻……………………………………………………………………………..23
表目錄
表1.1~1.3 線性模擬，真實自變數為常態分佈……………………………………24
表2.1~2.3 線性模擬，真實自變數為均勻分佈……………………………………25
表3.1~3.3 線性模擬，真實自變數為指數分佈……………………………………26
表4.1~4.3 線性模擬，真實自變數為卡方分佈……………………………………27
表5.1~5.3 二次迴歸模擬，真實自變數為常態分佈……………………….……..28
表6.1~6.3 二次迴歸模擬，真實自變數為均勻分佈……………………….……..29
表7.1~7.3 二次迴歸模擬，真實自變數為指數分佈……………………….……..30

[1]Amemiya and Fuller (1988). Estimation for the Nonlinear functional Relationship. The Annals of Statistics, Vol.12, No. 2. pp.497-509
[2]Carroll, R.J., Ruppert, D., Stefanski, L.A. (1995). Measurement Errors in Nonlinear Models. Chapman and Hall, London.
[3]C-L. Cheng and J.W.Van Ness (1999). Statistical Regression with Measurement Error. London: Arnold and New York: Oxford University Press. Kendall’s Library of Statistics 6.
[4]Fuller, W. A. (1987). Measurement Error Models. John wiley and Sons,New York.
[5]Kendall,M.G and Stuart, A. (1979). The Advanced Theory of Statistics, Charles Griffin; London.
[6]Nukamura, T. (1990). Corrected Scores Function for Errors-in Variables Models: Methodology and Application to Generalized Linear Models. Biometrika, 77,127-137.