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題 名 | 區域脊迴歸估計量的漸近性質研究(1)=Asymptotic Properties of the Local Ridge Regression Estimator(Ⅰ) |
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作 者 | 鄧文舜; | 書刊名 | 中國統計學報 |
卷 期 | 36:1 1998.03[民87.03] |
頁 次 | 頁29-41 |
分類號 | 319.51 |
關鍵詞 | 區域線性估計量; 區域脊迴歸估計量; 無母數迴歸; 隨機取樣; 脊迴歸; Local linear smoother; Local ridge regession estimator; Nonparametric regression; Random sampling; Ridge regession; |
語 文 | 中文(Chinese) |
中文摘要 | 在隨機取樣(random design)的無母數迴歸分析(nonparametric regression)中, 為了避開Fan(1993)的區域線性估計量(local linear smoother)會產生無限大的條件變異 數(conditional variance)的缺點,Seifert和Gasser(1996)將脊迴歸(ridge regression)觀 念結合Fan(1993)的區域線性估計量,構造出一個新的迴歸函數估計量,本文稱此估計量 為區域脊迴歸估計量(local ridge regression estimator)。然而他們沒有說明這個區域脊迴歸估 計量的理論性質。在迴歸函數具有連續二階導數的假設之下,我們研究該估計量的漸近性質 (asymptotic properties)。我們發現,若脊迴歸參數沒有經適當地選取,則所得到的迴歸函 數估計量的漸近偏差量(asymptotic bias)不會收斂到0。反之,若脊迴歸參數經適當地選取, 則所對應的區域脊迴歸估計量,在取樣點(design points)所發生區域的內部(interior),會 與Fan(1993)的區域線性估計量,兩者有相同的漸近偏差量及漸近變異數(asymptotic variance)。因此,在內部區域,脊迴歸的估計特性完全消失。另外,區域脊迴歸估計量將會 產生邊界效應(boundary effects),此即為區域脊迴歸估計量改善區域線性估計量,在條件 變異數方面的缺點,所付出的代價。在實務上如何選取帶寬值(bandwidth)以及脊迴歸參 數,本文考慮使用交叉有效法(cross-validation)。模擬研究顯示,在較小樣本規模(sample size) 之下,區域脊迴歸估計量可以比區域線性估計量有更小的樣本均方差(sample mean square error)。 |
英文摘要 | In the case of the random design nonparametric regression, to correct for the infinite conditional variance of the local linear smoother in Fan (1993). Seifert and Gasser (1996) apply the idea of ridge regression to the local linear smoother, and construct a new regression function estimator. The new estimator is called the local ridge regression estimator in this paper. However, the theoretical properties of the new estimator are not addressed. In this paper, we study the asymptotic properties of this new estimator. It shows that if the ridge regression parameters are not selected properly, then the resulting estimator is biased. On the other hand, if the ridge regression parameters are selected properly, then the resulting estimator and the local linear smoother have the same asymptotic bias and variance in the interior region. But, the local ridge regression estimator will suffer from boundary effects. This is the price paid by the local ridge regression estimator for correcting for the infinite conditional variance of the local linear smoother. To select both the bandwidth and the ridge regression parameters in practice, we suggest using the idea of cross- validation. Simulation studies demonstrate that the local ridge regression estimator could have smaller sample mean square error than the local linear smoother in small sample sizes. |
本系統中英文摘要資訊取自各篇刊載內容。