本篇論文研究由Bairamov以及Kotz (2002)提出的廣義Farlie-Gumbel-Morgenstern (FGM)聯結函數(copula)的數學與統計性質。首先，論文的第一部分回顧在廣義FGM聯結函數下，一些相關係數(dependence measure)的性質，其中包括斯皮爾曼相關係數(Spearman’s rho)、肯德爾相關係數(Kendall’s tau)、Kochar and Gupta’s dependence measure以及布萊斯特係數(Blest’s coefficient)，我們推導出了一些相關係數之間的關係、布萊斯特係數以及化簡過去文獻裡Kochar and Gupta’s dependence measure的結果。論文的第二部分主要探討在廣義FGM聯結函數下的相依競爭風險(dependent competing risks)分析，我們推導出了在FGM聯結函數模型下sub-distribution function的表示法，這是過去文獻中沒有討論過的，我們也證明我們的表示法在Burr III邊際分配下推廣了先前由Domma以及Giordano (2013)所提出的可靠度係數(reliability measure)。我們針對此篇論文提出的相依競爭風險模型使用最大概似估計法(Maximum likelihood estimation)來做參數估計，其中使用了隨機牛頓–拉弗森演算法(Randomized Newton-Raphson Algorithm)來最大化概似函數，我們設計了模擬研究來確認本篇論文模型以及方法的正確性，最後使用一組真實資料來做分析。;The thesis studies mathematical and statistical properties of the generalized Farlie-Gumbel-Morgenstern (FGM) copula (Bairamov and Kotz 2002). The first part of the thesis reviews several properties of dependence measures (Spearman’s rho, Kendall’s tau, Kochar and Gupta’s dependence measure, and Blest’s coefficient) under the generalized FGM copula. We give a few remarks on the relationship among the dependence measures, derive Blest’s coefficient, and suggest simplifying the previously obtained expression of Kochar and Gupta’s dependence measure. The second part of the thesis considers dependent competing risks analysis under the generalized FGM copula model. We obtain the expression of sub-distribution functions under the generalized FGM copula model, which has not been discussed in the literature. With the Burr III margins, we show that our expression has a closed form and generalizes the reliability measure previously obtained by Domma and Giordano (2013). We develop maximum likelihood estimation under the proposed competing risks models with a randomized Newton-Raphson algorithm for numerical maximization. We conduct simulations to check the correctness of our method and analyze a real dataset for illustration.
