皮爾森相關係數是一個計算成對資料 X, Y 線性相關程度常用的指標， Yang (2016) 根據 Jones (1996) 的局部相依函數 (local dependence function)提出了變化局部相關函數(varying local correlation function)，可以計算在一解釋變數 T=t0 附近時，變數 X 與 Y 的局部相關性，但只針對 X 與 Y 是成對資料的情形。我們想推廣前者至兩變數不需成對觀察，樣本數也不需要相同的情況。我們使用局部線性迴歸對兩變數進行曲線估計，並在格點上計算局部相關性，推廣出能適用在此資料型態的廣義局部相關函數估計式。本論文中探討了此估計式的漸近性質，並與變化局部相關函數的性質進行比較，最後以模擬來驗證理論結果，搭配實際資料分析，討論此估計式的表現與實際應用的成效。

The Pearson correlation coefficient is a commonly-used indicator of the degree of linear correlation of paired observations $(X_i, Y_i)_{i=1, \ldots, n}$. Based on the local dependence function by Jones (1996), Yang (2016) proposed the varying local correlation function which is a measure for the local correlation of two variables X and Y conditioned on an explanatory variable T = t0 when X and Y are paired. We generalize the varying local correlation function to unpaired observations of two variables with different sample sizes. Under the nonparametric regression setting, we use the local linear regression to estimate the curves of two variables and calculate the local covariance at a set of grid points, while retaining the estimates of local variances by Yang (2016). The asymptotic properties of generalized local correlation function are discussed and compared with the properties of the varying local correlation function. Finally, the theoretical results are confirmed by simulations, and illustrated by a real data example.

1.緒論 1
2.背景介紹 3
3.廣義局部相關函數 18
4.模擬與實際資料分析 34
5.結論與後續研究 51

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