存活分析主要在研究事件的發生時間；傳統的存活分析並不考慮治癒者(或免疫者)的存在。若以失敗為事件，且造成失敗的可能原因不止一種，但它們不會同時發生，則這些失敗原因就是失敗事件的競爭風險。競爭風險可分為有參數的競爭風險與無母數的競爭風險。本文同時考慮了有治癒與有參數的混合廣義Gamma分配，並將預估計的位置參數與失敗機率有關的參數與解釋變數結合，代入Choi及Zhou(2002)提出的最大概似估計量的大樣本性質。並考慮在治癒情況下，利用電腦模擬來估計在型一設限及無訊息(non-informative)的隨機設限(random censoring)下之一個失敗原因與兩個失敗原因下的參數平均數與標準差。 / The purpose of survival analysis is aiming to analyze the timeline of events. The typically method of survival analysis don’t take account of the curer (or the immune). If the event is related to failure and there are more than one possible reason causing the failure but are not happening at the same time, we called the possible reasons a competing risk for failed occurrence. competing risk can be categorized as parameter and non-parameter. This research has considered the generalized gamma distribution over both cure and parameter aspects. In addition, it combines anticipated parameter with covariate which affected to the possibilities of failure. Follow by the previous data, it is then substituted by the large-sample property of the maximum likelihood estimator which is presented by Choi and Zhou in 2002. With considering the possibilities of cure, it uses computer modeling to investigate that under the condition of type-1 censoring and non-informatively random censoring, we will find out the parameter mean and standard error that is resulted by one and two reason causes failure.

房屋貸款借款人對於其所負貸款債務的處分有兩種潛在風險行為，分別是提前清償及違約。這兩種借款人風險行為不管是對金融機構的資產管理，或是對近年在財務金融領域的不動產證券化而言，都是相當重要的探討議題，原因在於提前清償及違約帶來了利息收益與現金流量的不確定性，進而影響不動產抵押債權的價值。也就是為貸款承作機構、證券化保證機構及證券投資人帶來風險。 借款人決定提前清償及違約與否，除了與借款人自身特性及貸款條件有關外，尚受到隨時間經過而不斷變動的變數所影響，亦即許多影響因子並非維持在貸款起始點的狀態，而是會在貸款存續期間動態調整。進一步影響借款人行為，而這類變數即為時間相依變數(time –dependent variables，或time-varying variables)。因此，本研究利用便於處理時間相依變數的比例危險模型(Proportional Hazard Model)來分析借款人提前清償及違約風險行為，觀察借款人特徵、房屋型態、貸款條件及總體經濟等變數與借款人風險行為的關係。 實證結果顯示，借款人特徵部分的教育程度對提前清償及違約風險影響最為明顯，教育程度越高，越會提前清償，越低則較會違約。房屋型態則透天厝較非透天厝容易提前清償及違約。貸款條件中的貸款金額及貸款成數皆與違約為正相關，亦即利息負擔越重，借款人違約風險升高。總體經濟方面，借款人對利率變動最為敏感，反映利率代表借款人的資金成本，是驅動借款人提前清償及違約的財務動機與誘因。

競爭風險之下對邊際分配的估計，是許多領域中常遇到的問題。由於主要事件及次要事件互相競爭，只要一種事件先發生即終止對另一事件的觀察，在兩事件同時發生的機率為0之下，連一筆完整的資料我們都無法蒐集到。除非兩事件互為獨立或加上其它條件，否則會有邊際分配無法識別的問題。但是獨立的條件在有些情況下並不合理，為解決相依競爭風險之邊際分配無法識別的問題，可先假定兩事件發生時間之間的關係。 由於關聯結構定義出兩變數間的結合關係，我們可利用關聯結構解釋兩事件發生時間之間的關係。假定兩變數之相關性參數為已知，且採用機率積分轉換的觀念，本論文討論了Zheng 與 Klein提出的關聯結構-圖形估計量，是否會依設限程度、相關性強度和關聯結構形式的不同，以致估計能力有別。 / The problem of estimating marginal distributions in a competing risks study is often met in scientific fields. Because main event and secondary event compete with each other, and a first occurring event prevents us from observing another event promptly, the intact lifetimes or survival times are unable to be collected in the circumstances that the probability of both lifetimes coinciding is 0. Unless lifetimes being independent or adding other conditions, there is a problem that the marginal distributions are non-identifiable. But the condition of independence is not always reasonable, we may assume the relation between lifetimes has some special form Because the copula defines the association between two variables, it can be employed to explain relation between lifetimes. Assuming that the dependence parameter in the copula framework is known, and adopting the concept of the probability integral transformations, this thesis has demonstrated whether the estimating abilities of the copula-graphic estimator, that Zheng and Klein put forward, are different in rates of censoring, intensities of dependence, and forms of the copula.

當造成失敗的原因不只一種時，若各對象同一時間最多只經歷一種失敗原因，則這些失敗原因稱為競爭風險。然而，有些個體不會失敗或者經過治療之後已痊癒，我們稱這部分的群體為治癒群。本文考慮同時處理競爭風險及治癒率的混合模式，即競爭風險的治癒率模式，亦將解釋變數結合到治癒率、競爭風險的條件失敗機率，或未治癒下競爭風險的條件存活函數中，並以建立在完整資料上之擴充的概似函數為貝氏分析的架構。對於右設限對象則以插補方式決定是否會治癒或會因何種風險而失敗，並推導各參數的完全條件後驗分配及其性質。由於邊際後驗分配的數學形式無法明確呈現，再加上需對右設限者判斷其狀態，所以採用屬於馬可夫鏈蒙地卡羅法的Gibbs抽樣法及適應性拒絕抽樣法(adaptive rejection sampling) ，執行參數之模擬抽樣及設算右設限者之治癒或失敗狀態。實證部分，我們分析Klein and Moeschberger (1997)書中骨髓移植後的血癌病患的資料，並用不同模式之下的參數模擬值計算各對象之條件預測指標(CPO)，換算成各模式的對數擬邊際概似函數值(LPML)，比較不同模式的優劣。 / In case that there are more than one possible failure types, if each subject experiences at most one failure type at one time, then these failure types are called competing risks. Moreover, some subjects have been cured or are immune so they never fail, then they are called the cured ones. This dissertation discusses several mixture models containing competing risks and cure rate. Furthermore, covariates are associated with cure rate, conditional failure rate of each risk, or conditional survival function of each risk, and we propose the Bayesian procedure based on the augmented likelihood function of complete data. For right censored subjects, we make use of imputation to determine whether they were cured or failed by which risk and derive full conditional posterior distributions. Since all marginal posterior distributions don’t have closed forms and right censored subjects need to be identified their statuses, we take Gibbs sampling and adaptive rejection sampling of Markov chain Monte Carlo method to simulate parameter values. We illustrate how to conduct Bayesian analysis by using the bone marrow transplant data from the book written by Klein and Moeschberger (1997). To do model selection, we compute the conditional predictive ordinate(CPO) for every subject under each model, then the goodness is determined by the comparing the value of log of pseudo marginal likelihood (LMPL) of each model.