本論文主要探討核密度估計方法下距離相關係數的應用，並對其進行模擬與實際數據驗證。距離相關係數是一種用於衡量兩組多維隨機變數之間相關性的統計指標。傳統的皮爾森相關係數只能檢測線性關係，而距離相關係數則能夠檢測到更廣泛的非線性相關。核密度估計是一種無母數曲線估計方法，通過選取欲估計點周圍的樣本進行曲線配適，距離離估計點越近的樣本權重越大。核密度估計的核心概念在於利用加權過後的樣本來估計機率密度函數。本論文使用高斯核函數進行估計，並將其結果用於估計變數的聯合機率密度函數及邊際機率密度函數，從而得到估計後的聯合特徵函數及邊際特徵函數。本論文首先對距離相關係數進行了理論探討，並定義了相關的數學公式。接著，進行了一系列模擬實驗，包括生成高度相關及低度相關的數據，並計算出核密度估計距離相關係數、樣本距離相關係數與皮爾森相關係數進行比較。模擬結果顯示，與距離相關係數的統計指標在檢測非線性相關方面具有優越性。最後，將核密度估計距離相關係數應用於實際數據，驗證其在不同資料集上的適用性。

This thesis primarily explores the application of the distance correlation coefficient for kernel density estimators (KDE), with simulations and real data applications. The distance correlation coefficient is a statistical measure used to assess the association between two sets of multidimensional random variables. Unlike the traditional Pearson correlation coefficient, which can only detect linear relationships, the distance correlation coefficient can identify a broader range of nonlinear associations. KDE is a non-parametric density estimation method that fits a curve using data around the point to be estimated, with closer data points receiving more weight. This study employs Gaussian kernel functions for estimation and uses the results to estimate the joint and marginal probability density functions of the variables, thereby obtaining the estimated joint characteristic function and marginal characteristic function. The thesis first theoretically reviews the distance correlation coefficient and defines the relevant mathematical notation. Subsequently, a series of simulation experiments are conducted, including generating data with high and low correlations, and calculating the KDE-based distance correlation coefficients, the sample distance correlation coefficients, and the Pearson correlation coefficients for comparison. The simulation results demonstrate that the distance correlation coefficient is superior in detecting nonlinear correlations. Finally, the KDE-based distance correlation coefficient is applied to mortality data to verify its applicability across different datasets. (The Chinese abstract was written by myself and translated into English version using ChatGPT. Both my professor and I have revised the translated abstract.)

目錄
1 介紹 4
1.1 前言 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 研究動機 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 背景、定義及定理 5
2.1 核函數估計 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 距離相關係數 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 方法 9
3.1 KDE估計性質與其特徵函數 . . . . . . . . . . . . . . . . . . . . . . 9
3.2 利用二維核密度估計計算的距離共變異數 . . . . . . . . . . . . . . 12
4 模擬 15
4.1 模擬步驟介紹 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 模擬結果比較 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 統計量的表現 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 實際數據應用 52
6 結論 60
參考文獻 61

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