有測量誤差下的二次迴歸參數估計

The parameter estimation of quadratic polynomial model with measurement errors.

蕭富源(Fu-Yuan Xiao)

指導教授 ： 伍志祥

淡江大學/理學院/數學學系碩士班/碩士(2006年)

https://doi.org/10.6846/TKU.2006.00927

摘要

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。

關鍵字

測量誤差 ； 不可辨認性 ； 線性模式 ； 二次迴歸模型

並列摘要

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.

並列關鍵字

measurement error ； unidentifiability ； linear model ； quadratic polynomial model

參考文獻

[1]Amemiya and Fuller (1988). Estimation for the Nonlinear functional Relationship. The Annals of Statistics, Vol.12, No. 2. pp.497-509

[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.

[2]Carroll, R.J., Ruppert, D., Stefanski, L.A. (1995). Measurement Errors in Nonlinear Models. Chapman and Hall, London.