在線性模型中，使用誤差平方和(error sum of squares) 之差來衡量複雜模型與簡單模型配適能力差距的F 檢定是選擇適當模型常用的方法之一，Huang and Chen (2008) [7] 曾將該檢定的概念推廣至局部多項式迴歸模型(local polynomial regression model)（參考Fan and Gijbels, 1996 [3]）中、建立局部與總體變異數分解、並提出新的F 檢定統計量以檢驗由局部多項式迴歸配適的模型函數相對於常數函數是否更為恰當。本論文將仿效Huang and Chen (2008) [7] 的手法，將此檢定架構推廣至多變量局部線性迴歸模型(multivariate local linear regression model)（參考Ruppert and Wand, 1994 [17]）中，建立局部與總體變異數分解，進而定義兩個F 統計量以協助解決兩個感興趣的檢定問題：(i) 模型函數是否為常數函數、(ii) d 維模型函數是否可簡化為 (d-1) 維之函數，接著以模擬測試這兩個檢定統計量在二元解釋變數情況下的型一錯誤(type I error) 與檢定力(power)，觀察其隨著樣本數、解釋變數相關性、帶寬(bandwidth)、以及拒絕訊號不同時的變化，也同時提出實際執行本論文檢定的計算方法，其中包含如何將乘積核函數(product kernel function) 進行正規化(normalization)，最後則會將這兩個檢定方法應用在美國波士頓地區房價資料的分析。

In linear models, it is common to test the difference between two nested models by measuring the difference of their error sums of squares and performing an F-test. Huang and Chen (2008) [7] have extended the structure of this F-test to local polynomial regression (LPR) models (see Fan and Gijbels, 1996 [3]), constructed local and global ANOVA decompositions for LPR models, and defined an F-statistic to test whether a model function fitted by LPR is significant. This thesis extends this F-test to multivariate local linear regression (MLLR) models (see Ruppert and Wand, 1994 [17]) by mimicking a similar framework proposed by Huang and Chen (2008) [7]. We establish local and global ANOVA decompositions for MLLR models, and define two F-statistics corresponding to the following two hypotheses: (i) whether a model function fitted by MLLR is significant, and (ii) whether a model function fitted by MLLR with covariates X_2,..., X_d is more appropriate than a model function fitted by MLLR with covariates X_1,..., X_d. In the bivariate case (d = 2), the type I error and power for these two F-tests are investigated by simulations under different settings of sample sizes, correlations of covariates, values of bandwidth, and signals of rejection, while practical issues of implementing these two F-tests are also discussed, including normalization for the product kernel function. At last, these two F-tests are applied to the analysis of Boston house-price data.

1 緒論 1
2 背景介紹 3
2.1 局部多項式迴歸. . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 局部多項式迴歸的變異數分析及檢定. . . . . . . . . . . . . . . . 4
2.3 多變量局部線性迴歸. . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Tukey 中位數與bagplot . . . . . . . . . . . . . . . . . . . . . 6
2.5 其他文獻回顧. . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5.1 可加性模型的GLR 檢定. . . . . . . . . . . . . . . . . . . . . 8
2.5.2 多變量無母數迴歸模型的誤差分布估計與模型架構檢定. . . . . . . 8
3 多變量局部線性迴歸的檢定方法 10
3.1 變異數分解. . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 漸近投影矩陣. . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 檢定模型函數是否為常數函數. . . . . . . . . . . . . . . . . . . 14
3.4 檢定模型函數是否為變數 X_2,..., X_d 之函數. . . . . . . . . . . 15
4 模擬 18
4.1 核函數正規化方法. . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 模擬一. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.1 檢定模型函數是否為常數函數的模擬結果. . . . . . . . . . . . . 22
4.2.2 檢定模型函數是否為單變量 X_1 函數的模擬結果. . . . . . . . . .24
4.3 模擬二. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3.1 檢定模型函數是否為常數函數的模擬結果. . . . . . . . . . . . . 26
4.3.2 檢定模型函數是否為單變量 X_1 函數的模擬結果. . . . . . . . . .27
5 實際資料分析 30
6 結論與後續研究 39
7 附錄 42
7.1 證明. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.1.1 性質3.1 證明. . . . . . . . . . . . . . . . . . . . . . . . . 42
7.1.2 定理3.2 證明. . . . . . . . . . . . . . . . . . . . . . . . . 43
7.1.3 定理3.3 證明. . . . . . . . . . . . . . . . . . . . . . . . . 43
7.1.4 定理3.4 證明. . . . . . . . . . . . . . . . . . . . . . . . . 51
7.1.5 定理3.5 證明. . . . . . . . . . . . . . . . . . . . . . . . . 54
7.2 模擬結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.2.1 模擬一. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.2.2 模擬二. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
參考文獻 72

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