淡江大學覺生紀念圖書館 (TKU Library)
進階搜尋


下載電子全文限經由淡江IP使用) 
系統識別號 U0002-0107200705042000
中文論文名稱 有測量誤差下的線性迴歸參數估計
英文論文名稱 The parameter estimation of linear regression model with measurement errors
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 95
學期 2
出版年 96
研究生中文姓名 陳俊傑
研究生英文姓名 Chun-Chieh Chen
學號 693150087
學位類別 碩士
語文別 中文
口試日期 2007-06-11
論文頁數 57頁
口試委員 指導教授-伍志祥
委員-張三奇
委員-伍志祥
委員-黃逸輝
中文關鍵字 測量誤差  不可辨認性  線性模式  估計量族群 
英文關鍵字 measurement error  unidentifiability  linear model  population of estimation 
學科別分類 學科別自然科學數學
中文摘要 迴歸分析(Regression analysis)主要是探討解釋變數與反應變數間的各種關係。但是在建構回歸分析時,有時會發現所得到的資料是由真實解釋變數和測量誤差所結合而成的觀測值,像這種具測量誤差的迴歸模式,我們稱為測量誤差模式〈measurement error model〉。
在探討測量誤差模式的參數估計問題時,會遇到的參數不可辨認的問題。因此文獻中探討參數估計問題時,是在額外假設條件下估計參數,若不要額外假設條件而要去除參數不可辨認的模型,可利用過量參數法來估計參數,這時需假設觀測到具測量誤差的解釋變數的分佈要為不對稱分佈。
本文要探討的是在不限制有測量誤差的解釋變數的分佈下估計參數的方法,首先探討Y跟e^(t*w)的共變異數關係,可以得到如下式子
G(t)=β*M_{W_i^*}^'(t)-β*(M_{δ_i}^'(t)/M_{δ_i}(t))*M_{W_i^*}(t)
根據這一式子,我們分別假設特定條件,例如:已知δ的分配且變異數已知、不知道δ的分配但是利用泰勒展開式來求解等等,從中找到參數β的理論解,並且利用動差法來求得理論解所對應的估計量族群,隨後利用電腦模擬來討論所得到的估計量族群的效果,並說明是否有一個最佳的估計方法,如果沒有,那是否會有在特定條件下的最佳估計方法。
英文摘要 Regression analysis mainly discusses the relation between the explains variable and reaction variable. But sometimes we find the data that is combined by true explained variable and measurement error when we constructed back to regression analysis. We call this kind of regression model as measurement error model.
When we discuss the problem of parameter estimate of measurement error model, the problem of parameter not recognizable will be meeting for us. Therefore literatures treats of the problem in the additional assumption condition. If don't suppose the additional assumption condition but we also want to dispose of the parameter not recognizable, we can estimate the parameter by overparameterization method, at this we need supposed the explains variable of having measurement error that must want to dissymmetrical density.
In this text, we discuss the method of estimating parameter that the density of explains variable of having measurement error is not abridge, we inquire the covariance of y and e^(t*w) and we can get a formula as follows
G(t)=β*M_{W_i^*}^'(t)-β*(M_{δ_i}^'(t)/M_{δ_i}(t))*M_{W_i^*}(t)
Now according to this formula, we suppose some particular conditions. For example: first we know the density of δ and the variance is known, second we are unknown the density of δ. Using Taylor expansion can fine the estimation. Then we want to fined the theoretical solution of parameter β.By using moment method, we can obtain the population of estimator. With the computer simulation, we could discuss the effect of population of estimation, and explain whether have a best estimate method. If it couldn’t, it would have the best estimation under the particular condition?
論文目次 目錄
第一章 緒論...............................................1
第二章 β的理論解..........................................3
2.1 Y跟e^(t*w)的共變異數..................................3
2.2 知道δ的分配-沒有未知參數時............................4
2.3 知道δ的分配-含有未知參數..............................5
2.4 不知道δ的分配.........................................6
2.4.1 逼近法...........................................7
2.4.2 比較係數法.......................................7
第三章 利用動差法求得β的估計量............................9
3.1 β的估計量.....................................10
第四章 電腦模擬與其結果..................................13
4.1 運用統計軟體的模擬條件........................13
4.2 模擬結果與討論................................14
4.2.1 當n = 25時...............................14
4.2.2 當n = 50時...............................15
4.2.3 當n = 100時..............................16
4.2.4 當n = 200時..............................18
4.2.5 當n = 400時..............................19
第五章 結論..............................................21
參考文獻.................................................24
附錄一...................................................25
圖1.1.............................................25
圖1.2.............................................26
圖1.3.............................................27
圖1.4.............................................28
圖1.5.............................................29
圖1.6.............................................30
圖1.7.............................................31
圖1.8.............................................32
圖1.9.............................................33
附錄二...................................................34
圖2.1.............................................34
圖2.2.............................................35
圖2.3.............................................36
圖2.4.............................................37
圖2.5.............................................38
圖2.6.............................................39
圖2.7.............................................40
圖2.8.............................................41
圖2.9.............................................42
附錄三...................................................43
圖3.1.............................................43
圖3.2.............................................44
圖3.3.............................................45
圖3.4.............................................46
圖3.5.............................................47
圖3.6.............................................48
圖3.7.............................................49
圖3.8.............................................50
圖3.9.............................................51
附錄四...................................................52
圖4.1.............................................52
圖4.2.............................................53
圖4.3.............................................54
圖4.4.............................................55
圖4.5.............................................56
圖4.6.............................................57
參考文獻 Amemiya and Fuller (1988). Estimation for the Nonlinear functional Relationship.The Annals of Statistics, Vol.12, No.2. pp.497-509

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

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.

Fuller, W.A. (1987). Measurement Error Models. John wiley and Sons,New York.

Kendall.M.G and Stuart, A. (1979). The Advanced Theory of Statistics, Charles Griffin; London.

Nakamura, T. (1990). Corrected Scores Function for Errors-in Variables Models: Methodology and Application to Generalized Linear Models. Biometrika. 77,127-137
論文使用權限
  • 同意紙本無償授權給館內讀者為學術之目的重製使用,於2007-07-05公開。
  • 同意授權瀏覽/列印電子全文服務,於2007-07-05起公開。


  • 若您有任何疑問,請與我們聯絡!
    圖書館: 請來電 (02)2621-5656 轉 2281 或 來信