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.
