傳統的統計方法中，在分析具有左截斷(left-truncated)的資料時常常會把感興趣的隨機變數與其左截斷隨機變數具獨立性的假設。但已有許多文章指出在真實資料中獨立性的假設並不合適。此篇論文中我們建模的方式是給定兩隨機變數的邊際分配並使用連結函數(copula function)來建構我們的聯合機率分配。用此全母數模型即可解決獨立性假設不合適的問題。在此種模型下我們對inclusion probability 推導出較為簡化的公式並給出在某些條件下此函數對參數微分的公式。在連結函數為Clayton copula 邊際分配皆為韋伯(Weibull)的模型下，我們推導出對數概似函數對參數的一次及二次微分並使用了隨機牛頓-拉弗森演算法(Randomized Newton-Raphson algorithm)得到參數的最大概似估計量(Maximum likelihood estimator)。在最後我們使用了一組煞車皮的壽命資料作為實例的分析。;Traditional statistical methods for left-truncated lifetime data rely on the independence assumption regarding the truncation variable. However, the dependence between a lifetime variable of interest and its left-truncation variable usually occurs in many real data from reliability and biomedical analysis. In this paper, we propose a copula-based dependence model between and with the marginal distributions specified by parametric models. Then we consider the maximum likelihood estimator (MLE) under the copula-based dependence model between and . To calculate the MLE of the unknown parameters , explicit formulas of the inclusion probability and its partial derivatives are obtained under the Clayton copula and Weibull marginal model, which are new results in this paper. Then we derive explicit expression for the randomized-Newton-Raphson algorithm for maximizing the log-likelihood. We perform simulations to verify the correctness of the proposed method. We illustrate our method by real data from a field reliability study on the lifetimes of brake pads.
