Estimation of umbrella shaped regression function

傘型迴歸函數是類似傘的形狀的迴歸函數，只要符合先上升後下降的趨勢皆為傘型迴歸函數。無母數迴歸函數中最常見的方法之一是樣條(Splines)迴歸函數。樣條為充分平滑分段多項式函數，而節點(knots)為平滑多項式函數連接的地方。在本論文中，將節點以等距離擺放並以AIC(Akaike information criterion)值得到合理的節點數。用三種方法的樣條迴歸函數去估計傘型函數。第一種為RSPL(restrictted spline regression)，也就是有形狀限制時的樣條迴歸函數。第二種是CSPL(concave spline regression)，是參考Meyer寫的樣條迴歸函數，此樣條迴歸函數為凹函數(concave function)。最後一種則稱SPL(spline regression)，為沒有形狀限制也不是凹函數的樣條函數。以IMSE為評估標準，IMSE越小，則代表此方法估計的越好。由模擬結果，在估計先上升後下降的函數時，用RSPL的方法去估計會得到最小的IMSE；而在估計凹函數時，則是CSPL會得到最小的IMSE。利用RSPL和SPL兩個方法估計由中央氣象局蒐集最近13年(1998-2010)的月均溫資料並探討最近幾年的月均溫資料趨勢是否有改變。未來假如需要估計傘型函數時，則可利用本篇所述的方法去估計。In this thesis, we consider the problem of estimating a regression function assuming the regression function is unimodal. The proposed method is to model the regression function as linear combination of B-spline basis functions with equally spaced knots, and the number of knots is determined using AIC (Akaike information criterion). Specific constraints are placed on the coefficients of basis functions to ensure that estimated regression function is unimodal. The coefficients are estimated using least square method. The proposed method is refered as RSPL and is compared with two other methods: SPL and CSPL, where SPL is similar to RSPL except that the coefficients of basis functions are estimated without any constraints, and CSPL gives concave regression function estimates. Simulation results show that RSPL outperforms SPL and CSPL when the true regression function is unimodal but not concave, and CSPL outperforms RSPL and SPL when the true regression function is concave. Also, RSPL is applied to temperature data to estimate temperature trend within one year