我們主要的研究內容為使用的共變數分析探討不同曲線下的統計檢定，也就是說在不同的因子下，探討曲線之間的關係。在給定的假設條件下，本論文中提到了五個統計檢定，並且推導出每個檢定中的SSR與SSE，進而產生F 檢定的統計量。在模擬的部分有相對應的結果和性質。

Contents
1 INTRODUCTION 1
2 BACKGROUND 3
2.1 Local polynomial regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Asymptotic projection matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Nonparametric analysis of covariance . . . . . . . . . . . . . . . . . . . . . . . 5
3 SEMIPARAMETRIC ANALYSIS OF COVARIANCE 10
3.1 ANOVA Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Semiparametric F-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 SIMULATION STUDY 17
5 DISCUSSION 21
FIGURES 27
APPENDIX 28
Conditions (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Proof of equation (3.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Proof of equation (3.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Proof of Conjecture 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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