在隨機取樣(random design)的無母數迴歸(nonparametric regression)分析中，區域線性估計量(local linear estimator)有許多較好的漸近性質，因此較為多數人所稱道；然而在有限樣本場合，若資料稀疏(sparse)或自變數之間有共線性(multicollinearity)的情形之下，區域線性估計量其條件變異數(conditional variance) 沒有上界(unbounded)。為了避免這些情形，因此有許多的改進方法被提出，如Seifert and Gasser(1996)提出的區域線性脊迴歸(local linear ridge regression)方法、Hall and Marron(1997)的縮小(shrinkage)方法以及Chu and Deng(2003)插補值(interpolation method)的區域線性方法。

    本文在討論這些區域線性估計量改進法的比較，並在合理的樣本之下，以交叉比對法(cross validation)來選取各平滑參數(smoothing parameter)，來判斷出在實務上，插補值的區域線性的改進法較能得到較低的樣本期望積分方差(sample mean integrated square error)，並且在實務上所得出的曲線或曲面也較平滑。

In the case of the random design nonparametric regression, local linear estimator are an attractive choice due to many asymptotic properties. For the local linear estimator, however, if the data is in the presence of sparse and multicollinear design, it has the problem of unbounded finite sample variance. To overcome the problem, there are many remedies for proposing, such as that the method of the local linear ridge regression by Seifert and Gasser(1996), of shrinking by Hall and Marron(1997), and the interpolation method of local linear estimator by Chu and Deng(2003).

    In this paper, we will compare the above estimator. We also use cross validation idea to select their smoothing parameters to determine that the interpolation method of local linear estimator will get the lower sample mean integrated square error on the practice under the reasonable sample size, and it can get the smoother estimated curve and surface.

目錄

第１章　導論  1

1.1.	無母數迴歸分析簡介	   1

1.2.	區域線性估計量	2

1.3.	有限樣本下的區域線性估計量	7

1.4.	共線性樣本下的區域線性估計量	7


第２章　區域線性估計量一維空間改進方法及其模擬研究	11

2.1.	區域線性脊迴歸估計量	11

2.2.	縮小估計量	12

2.3.	插補值的區域線性估計量	13

2.4.	實務上參數的選取方法	14

2.5.	一維自變數資料有稀疏情形的模擬研究	15


第３章　區域線性估計量二維空間改進方法及其模擬研究	30

3.1.	區域線性脊迴歸估計量	30

3.2.	插補值的區域線性估計量	31

3.3.	二維自變數資料有共線情形的模擬研究	32


第４章  結論	39

4.1.	第二章模擬研究結果	39

4.2.	第三章模擬研究結果	40


參考文獻	41
 

圖目錄

圖1. 	迴歸函數m1(x)以最小化積分方差值得到之mˆLLRRE(x)、mˆS(x)和mˆFILLE(x)                                      18
圖2.	迴歸函數m2(x)以最小化積分方差值得到之m^LLRRE(x)、m^S(x)
和m^FILLE(x)                                                   19
圖3.	迴歸函數m3(x)以最小化積分方差值得到之m^LLRRE(x)、m^S(x)
和m^FILLE(x)                                                   20
圖4.	迴歸函數m1(x)以最小化交叉比對值得到之m^LLRRE(x)、m^S(x)
和m^FILLE(x)                                                   21
圖5.	迴歸函數m2(x)以最小化交叉比對值得到之m^LLRRE(x)、m^S(x)
和m^FILLE(x)                                                   22
圖6.	迴歸函數m3(x)以最小化交叉比對值得到之m^LLRRE(x)、m^S(x)
和m^FILLE(x)                                                   23
圖7.	迴歸函數m1(x)以最小化交叉比對值所得的之m^LLRRE(x)、m^S(x)
和m^FILLE(x)在101個點上的估計值的平均數與標準差。		   25
圖8.	迴歸函數m2(x)以最小化交叉比對值所得的之m^LLRRE(x)、m^S(x)
和m^FILLE(x)在101個點上的估計值的平均數與標準差。        	   26
圖9.	迴歸函數m3(x)以最小化交叉比對值所得的之m^LLRRE(x)、m^S(x)
和m^FILLE(x)在101個點上的估計值的平均數與標準差。　　　　　　   　27
圖10.	迴歸函數m(x1,x2)和以最小化積分方差值得到之m^LLE(x1,x2)
、m^LLRRE(x1,x2)和m^FILLE(x1,x2) 　　　　　			　			   34
圖11.	迴歸函數m(x1,x2)和以最小化交叉比對值得到之m^LLE(x1,x2)
、m^LLRRE(x1,x2)和m^FILLE(x1,x2) 　　　　　　					   36
 
表目錄

表1.	m^NW(x)、n^GM(x)和m^LLE(x)的漸近變異數和漸近偏誤。　          　6
表2.	m^LLRRE(x)、m^S(x)和m^FILLE(x)以最小化期望積分方差所得之計算結果。                                                        28
表3.	m^LLRRE(x)、m^S(x)和m^FILLE(x)以最小化交叉比對值所得之計算結果。	                                                       29
表4.	m^LLE(x1,x2)、m^LLRRE(x1,x2)和m^FILLE(x1,x2)以最小化期望積分方差所得之計算結果。                                              38

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