Advancements in Cure Rate Modelling and Analysis of Multi-state Coherent Systems

在过去的六年里，我的研究主要集中在生存分析、可靠性分析、分布理论、有序数据分析和随机排序等领域。在未来的五年里，我计划在所有这些领域中追求一些突出的问题。生存分析：由于包括癌症和心脏病在内的一些疾病的治疗取得了很大的进步，一定比例的患者对治疗反应良好，可能在很长一段时间内不会复发。他们被称为“治愈者”或“长期幸存者”，意思是他们已达到疾病无法检测的健康阶段，并且在很长一段时间内不会复发。为了估计这一治愈率，文献中讨论了许多灵活的治愈率模型和相关的推理方法。在这个项目中，我计划开发一种新型的多阶段破坏性治疗模型，以模拟可能在多次发生的癌细胞破坏，例如在每次化疗或放疗后。然后，我计划为这种多阶段破坏性治愈模型开发推理方法。可靠性分析：使用“系统签名”的概念对相干系统寿命数据的分析进行了广泛的研究。利用签名对系统和部件的寿命特性开发了有效的推理方法。但是，这些发展大多是基于两国体制（或上或下）。在这里，我计划考虑具有二元成分的多态相干系统，并研究相应的系统签名、有序系统签名及其性质。然后，我计划使用它们来开发推理方法。分布理论：文献中研究了几种二元和多元离散分布。我计划使用COM-Poisson模型，一般二元泊松模型和复合来开发灵活的多元离散分布，并开发它们的推理方法。有序数据分析和随机排序：近年来，人们讨论了比例风险、比例逆转风险和比例族的最小值和最大值的各种随机排序，以及它们在精算科学索赔金额中的应用。我计划将这些结果扩展到一般阶统计量，并推广到相关情况。对该领域和加拿大的预期结果和利益：我指导的大多数博士后和博士学生都在许多大学任职，包括加拿大的卡尔加里、马尼托巴、滑铁卢、渥太华和麦克马斯特，以及美国的普渡、锡拉丘兹、南卫理公会、明尼苏达州和德克萨斯州。还有一些人加入了加拿大公司，如庞巴迪、蒙特利尔银行、丰业银行、加拿大商业银行、道明银行、通用电气金融、罗杰斯和加拿大轮胎。我预计拟议的研究工作将产生类似的结果和效益。

During the past six years, my research has primarily focused in the areas of Survival Analysis, Reliability Analysis,  Distribution Theory, Ordered Data Analysis and Stochastic Orderings. During  the next five years, I plan to pursue some outstanding problems in all these areas. SURVIVAL ANALYSIS: Due to the great advancements in the treatments of some diseases including  cancer and heart disease, a certain proportion of patients respond favourably to a treatment and may not show recurrence for a very long period of time.  They are referred to as "cured individuals" or "long-term survivors" meaning that they have reached a health stage wherein the disease is undetectable and would not recur  over a long  period.  For estimating this cure proportion, many flexible cure rate models and associated  inferential methods have been discussed in the literature.  In this project, I plan to develop a novel class of multi-stage destructive cure models to  model the possible destruction of cancerous cells that can happen at multiple times, for example, after the administration of each chemotherapy or radiation treatment.  I then plan to develop  inferential methods for such a multi-stage destructive cure model. RELIABILITY ANALYSIS:  Analysis of coherent system lifetime data has been studied extensively using the concept of "system signature". Efficient  inferential methods have  been developed for lifetime characteristics of systems and of components using signatures. But, most of these developments are on two-state systems (either up or down). Here, I plan to consider multi-state coherent systems with binary components and study the corresponding system signatures, ordered system signatures and their properties. I then plan to use them  to develop  inferential methods. DISTRIBUTION THEORY: Several bivariate and multivariate discrete distributions have been studied in the literature.   I plan to use the COM-Poisson model, general bivariate Poisson model and compounding to develop flexible  multivariate discrete distributions and also develop inferential methods for them. ORDERED DATA ANALYSIS AND STOCHASTIC ORDERINGS:  Recently various stochastic orderings have been discussed for minima and maxima from Proportional Hazards, Proportional Reversed Hazards and Scale families, and their application to claim amounts in Actuarial Science.  I plan to extend these results to general order statistics and also to generalize to dependent case. Anticipated Outcomes and Benefits to the Field and to Canada: Most post-docs and PhD students I supervised have taken up positions in many Universities, including Calgary, Manitoba, Waterloo, Ottawa and McMaster in Canada, and Purdue, Syracuse, Southern Methodist, Minnesota,  and Texas in USA.  Some others have joined Canadian companies such as Bombardier, BMO, Scotia Bank, CIBC, TD Bank, GE Capital, Rogers and Canadian Tire.  I anticipate the proposed research work to result in  similar outcomes and benefits.