觀察患有某一疾病的個體，許多的醫學研究表明，接受適當的治療後，有一比例的個體會治癒、不再復發或者死於其他疾病，當個體不會發生所要探討的疾病事件，我們稱其為治癒個體。治愈模型可用來描述治癒個體並配合著適當的解釋變數來預測治癒率，有界失效治癒模式為傳統用來預測治癒率的兩類模型之一，我們將該模型在描述治癒率的部分從有母數建模拓展到無母數建模，而模型中的基準函數仍維持有母數分布的假設。我們透過局部多項式回歸的手法藉由最大局部概似函數來求得模型中治癒率部分的參數，並提出兩步驟演算法，迭代求得基準函數的參數與描述治癒率部分的參數。在理論部分，我們目前推導出讓最大概似函數滿足凹向性的條件，最後我們將所提的模型應用到兩份模擬資料與兩份實際資料。

Many medical studies show that there are subjects who are cured, free of disease, or die of other causes after treatments. Those subjects come from the nonsusceptible population of the disease and cure models may be used to predict the cure rates with suitable covariates. In
our thesis, we extend bounded cumulative hazards models, which is one of the two main types of cure models, to a nonparametric setting for estimating the cure rates and assume that the baseline function in the model follows a parametric distribution. We adopt the local polynomial
approach and use the local likelihood criterion to derive estimators of cure rates. This way we adopt a flexible method to estimate the cure rate, the important part in cure models, and a convenient way to model the baseline function, which is less useful in practice. We also derive the convex conditions for the extended cure model. We simulate two examples to examine the performance of our proposed methods. Finally, we apply the extended model to two real datasets to predict cure rates with a continuous covariate.

1 Introduction
2 Background
2.1 Cure rate model
2.1.1 Two-Component Mixture Model
2.1.2 Bounded Cumulative Hazard Model
2.2 Local Polynomial Modelling and Local Likelihood
2.2.1 Local Polynomial Modelling
2.2.2 Local Likelihood Estimation
3 Methodology
3.1 Data Structure
3.2 Semi-parametric Cure Rate Model – Local Polynomial Bounded Cumulative HazardModel
3.2.1 Estimatin Equation
3.2.2 Two-Step Estimation
3.3 Convex Property
4 Simulation Results
4.1 Example 1
4.2 Example 2
5 Real Data Analysis
5.1 Bone Marrow Transplantation for Leukemia
5.2 Death Time of Kidney Transplant Patients
6 Discussion and Future Work
Appendix A
Reference

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