對於單調迴歸函數，尋求一個簡單、平滑及有效的估計是受到相當大關注。在本論文中，我們使用伯氏多項式來模型化迴歸函數，並以最小平方法來來估計單調迴歸函數。我們使用交叉驗證法來決定伯氏多項式的階數，提出一個以懲罰函數法為原理之演算法來計算所提估計，並提供迴歸函數之單點信賴帶的估計。模擬試驗及實際資料分析說明了此統計方法的可行性。

Search for a simple, smooth and efficient estimate of a smooth monotone regression function is of considerable interest. In this thesis, we describe a least square method for monotone regression in which the regression function is modeled by the Bernstein polynomial. We employ the cross-validation criterion to determine the degree of Bernstein polynomial, propose a penalty function method based algorithm to compute estimate and provide a pointwise confidence band for regression function. The success of this method is demonstrated in simulation studies and in an analysis of real data.

目錄
第一節 緒論............................1
第二節 估計方法........................3
       2.1 參數a與σ的估計方法..........3
       2.2 E{hat(fa(x))}信賴區間估計...4   
       2.3 伯氏多項式階數J的選取.......5
第三節 演算法..........................7
第四節 模擬試驗.......................11
       4.1 估計方法的數值表現.........11
       4.2 估計方法的穩健性...........13
第五節 實例分析.......................21
第六節 討論...........................23
參考文獻..............................24
附錄..................................25

圖目錄
圖1　真實迴歸函數為5階伯氏多項式fa(x)係數a=       
     (1.0,1.1,1.2,7.0,7.1,7.3)'，n=100，σ^2=1，1000組樣本的
     模擬結果..................................17
圖2　真實迴歸函數為5階伯氏多項式fa(x)係數a=
     (1.0,1.1,1.2,7.0,7.1,7.3)'，n=100，σ^2=1，1組樣本的模
     擬結果..................................17
圖3　真實迴歸函數為非多項式f(x)=ex+tan(2.3x-1)+1.5，n=100，
     σ^2=1，1000組樣本的模擬結果.............. 20
圖4　真實迴歸函數為非多項式f(x)=ex+tan(2.3x-1)+1.5，n=100，
     σ^2=1，1組樣本的模擬結果.............. 20
圖5　Mammen et al. (2001)方法所得之迴歸函數估計.....22
圖6　根據本論文所提最小平方法所得之迴歸函數估計及其95%信賴區
     間.........22

表目錄
表1　真實迴歸函數為5階伯氏多項式fa(x)係數a=
     (1.0,1.1,1.2,7.0,7.1,7.3)'之模擬試驗......15
表2　真實迴歸函數為5階伯氏多項式fa(x)係數a=
     (1.0,1.1,1.2,7.0,7.1,7.3)'在hat(fa(tilta(x50)))之數值表
     現.....16
表3　真實迴歸函數為非多項式f(x)=ex+tan(2.3x-1))+1.5之模擬試
     驗.....18
表4　真實迴歸函數為非多項式f(x)=ex+tan(2.3x-1))+1.5在hat(fa
     (tilta(x50)))之數值表現.....19

[1]Chang, I.S., Chien, L.C., Hsiung, C.A., Wen, C.C. and Wu, Y.J. (2007) Shape restricted regression with random Bernstein polynomials. In R. Liu, W. Strawderman and C.H. Zhang (eds), Complex Dataset and Inverse Problems. IMS Lecture Notes – Monograph Series. 54 187-202.
[2]Hastie, T.J., and Tibshirani, R.J. (1990) Generalized additive models. London: Chapman & Hall.
[3]Mammen, E., Marron, J.S., Turlach, B.A., and Wand, M.P. (2001) A general projection framework for constrained smoothing. Statistical Science, 16 232–248.
[4]Takezawa, K. (2006) Introduction to nonparametric regression. Hoboken New Jersey: Wiley-Interscience.
[5]Yang, W.Y., Cao, W., Chung, T.S., and Morris, J. (2005) Applied numerical methods using MATLAB. Hoboken New Jersey: Wiley-Interscience.