臺灣博碩士論文加值系統

用最小平方法(LSE)配適線性迴歸模型是一般常見的估計方法，但是並不是每種資料型態都適合使用，因為線性模型中自變數(independent variable)或共變數(covariate)其資料來源有些是 deterministic 而有些是收集而來且具有分配的隨機變數值。因此當模型中的自變數是具有分配的隨機變數值時，使用最小平方法去估計參數可能會出現一些不合適的情況。為了改善這些缺點本文提出另一個新方法，首先藉由用convolution的方式去推導 Y 的邊際機率密度函數進而獲得Likelihood Function，但是在推導 Y 的邊際機率密度函數與 Likelihood Function 的過程須用到Monte Carlo Method來逼近算式，然後再藉由Newton-Raphson Method 來估計 β 參數值。由於為了使Likelihood的值在現有給予的資料上盡可能靠近最佳值，而使Newton-Raphson Method 估計出的 β 參數值更準確，所以在資料上做了特殊選取，最後藉由電腦模擬比較新方法與最小平方法估計結果。

In linear regression modeling, the method of least squares is a general way to find the optimal linear relation of a dependent variable and multiple independent variables (covariates) provided that the covariates are assumed to be given or deterministic to the model. In practice, the covariates can be collected from real data sources and by natural follow some distributions. The ordinary least square estimates can be less efficient if the covariates are stochastic. In this study, we propose a new method to estimate the regression. We estimate the parameters by maximizing the integrated likelihood function, that is, the joint marginal distribution of the dependent variable. We approximate the integrated likelihood function using selected Monte Carlo samples of covariates through that only important probability weights are accumulated in the likelihood function. The maximum likelihood estimation is obtained applying the Newton-Raphson iterations on the approximated likelihood function. Simulation examples are given and the results are compared to the least squares estimates.

目錄
摘要 i
誌 謝 iii
目錄 iv
第一章 緒論 1
第二章 研究目的與方法 5
第三章 模型架構與推導 7
第一節 模型介紹 7
第二節 最大概似函數Likelihood 7
第三節 σ估計式 11
第四章 樣本選取 13
第五章 邊際值討論 16
第六章 結果比較 19
第七章 結論 22
附錄 23
參考文獻 25

1.Anderson, E. C. (1999) Monte Carlo Methods and Importance Sampling. Lecture Notes for Stat 578C 　
　Statistical Genetics.
2.Chatterjee, S. and Hadi, A. S. (2012) Regression Analysis by Exmaple, fifth Edition. John Wiley & Sons,
　Inc.
3.Lai, T. L and Wei, C. Z. (1982) Least Squares Estimates in Stochastic Regression Models with Applications
　to Identification and Control of Dynamic Systems. Annals of Statistics, Vol. 10, No.1, 154-166
4.Parzen, M., Lipsitz S. R., Ibrahim, J. G. and Lipshultz, S. (2002) A Weighted Estimating Equation for
　Linear Regression with Missing Covariate Data. Statist. Med. 21, 2421-2436.
5.Sprott, D. A. (1983) Estimating the Parameters of a Convolution by Maximum Likelihood. Journal of the
　American Statistical Association, Vol.78, NO. 382, 457-460.
6.Sclove, S. L. and Van Ryzin, J. (1969) Estimating the Parameters of Convolution. Journal of the Royal
　Statistical B, Vol. 31, No. 1, 181-191.