我們提出以半母數混合模型(Semiparametric Model)來分析死亡時間(Failure Time)和治癒時間(Cure Time)的混合資料。在這模型中我們使用無母數最大概然估計量(Nonparametric Maximum Likelihood Estimate (NPMLE))來估計死亡率(Case Fatality Rate)、回歸參數(Regression Parameter)和累積風險函數(Cumulative Hazard Function)。首先、在適當的條件下建立模型參數的Identifiability。接著證明最大概然估計量的存在性並說明其為一積分方程之解。利用此積分方程及Empirical Process Theory，證得其漸近一致性(Asymptotic Consistency)。 接著，我們利用我們所得到的積分方程和得分函數(Score Function)提出一個演算法來計算無母數最大概然估計量。使用此演算法來做統計模擬，得到了令人滿意的結果，使我們確認此模型和演算法的適切性。最後，我們使用這個模型來分析台灣疾病管制局(CDC Taiwan)所提供嚴重急性呼吸道症候群(SARS)的資料。經由我們的演算法對這些資料計算所得到的結果，對台灣嚴重急性呼吸道症候群這個傳染病建構了一個應用範圍廣泛的成果。 In this paper, we study nonparametric maximum likelihood estimators (NPMLE) in a semiparametric mixture model for cure time and failure time. This model is motivated by the study of fatality rate, time from onset to discharge and time from onset to death for SARS (severe acute respiratory syndrome) patients. SARS patients are kept in isolation until recovery or death. Because of no known treatment or preventive measure, it is important to know the case fatality rate and the distribution of admission-to-death and admission-to-discharge for the study of transmission dynamics and for better planning of patient care capacity. The identifiability of the parameters, the existence of NPMLE, and their asymptotical consistency are established under certain regularity conditions. We also propose a self-consistency based algorithm for computing the nonparametric maximum likelihood estimates in this model. The performance of this method is successfully demonstrated in a simulation study and in the analysis of Taiwan SARS data.
