對於凹性迴歸函數，尋求一個簡單、平滑及有效的估計是受到相當大的關注。在本論文中，我們使用伯氏多項式來模型化迴歸函數，並以最小平方法來估計凹性迴歸函數。我們使用AIC驗證法來決定伯氏多項氏的階數，提出一個以懲罰函數為原理之演算法來計算所提的參數估計，並提供迴歸函數之單點信賴區間估計和預測區間帶。模擬試驗及實際分析說明了此統計方法的可行性。

Search for a simple, smooth and efficient estimator of a smooth concave regression function is of considerable interest. In this thesis, we describe a least square method for concave regression in which the regression function is modeled by the Bernstein polynomial. We employ the Akaike’s information criterion to determine the degree of Bernstein polynomial, propose a penalty function method based algorithm to compute estimate and provide a pointwise confidence interval estimator and a prediction interval 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 參數的估計方法..............................3
2.2 信賴區間估計與預測區間估計..........4                   
2.3 伯氏多項式階數J的決定..................5
第三節	  演算法.....................................6
第四節	  模擬試驗................................10
4.1 估計方法的數值表現.....................10
4.2 估計方法的穩健性........................12
第五節	  實例分析.................................14
第六節	  討論.......................................15
參考文獻............................................23
附錄..................................................24

圖目錄
圖1  真實函數為5階伯氏多項式f(x)係數a=(1,4,6,5,3,0.5)1組樣本的模擬結果......................................................20
圖2  真實迴歸函數為非多項式f(x)=4sin(3x)1組樣本的模擬結果.......................................................21
圖3  (a)為肥料在各單位的含氮量下之玉米的平均收穫量之折線圖及(b)為根據本論文所提之最小平方法所得之迴歸函數估計、其95%之信賴區間及95%預測區間.......................................................22

表目錄
表1  真實迴歸函數為5階伯氏多項式f(x)係數a=(1,4,6,5,3,0.5) 且由AIC驗證法決定伯氏多項式階數之模擬實驗....................................16
表1’ 真實迴歸函數為5階伯氏多項式f(x)係數a=(1,4,6,5,3,0.5) 且固定伯氏多項式階數J=10 之模擬實驗.............................................17
表2  真實迴歸函數為非多項式f(x)=4sin(3x)且由AIC驗證法決定伯氏多項式階數之模擬實驗......................................................18
表2’真實迴歸函數為非多項式f(x)=4sin(3x)且固定伯氏多項式階數J=10之模擬實驗.......................................................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]  Takezawa, K. (2006) Introduction to nonparametric regression. Hoboken New Jersey: Wiley-Interscience.

[4]  Yang,W.Y.,Cao, W.,Chung,T.S.,and Morris,J.(2005) Applied  numerical methods using MATLAB. Hoboken New Jersey: Wiley-Interscience.

[5]  Cheng, Y.I (2009) Least Square Estimation for Monotone Regression.
   Department of Mathematics,Tamkang university,master thesis.

[6]  Hildreth, C. (1954) Point estimates of ordinates of concave function.