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.
