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、随机分布理论、有序数据分析和随机排序等领域。在接下来的五年里，我计划在所有这些领域都追查一些突出问题。生存分析：由于癌症和心脏病等疾病的治疗取得了很大进步，一定比例的患者对治疗反应良好，可能会在很长一段时间内不会复发。他们被称为“治愈的个体”或“长期幸存者”，意思是他们已经达到了疾病无法检测到并且不会在很长一段时间内复发的健康阶段。为了估计这个治愈比例，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，我计划开发一类新的多阶段破坏性治疗模型，以模拟可能多次发生的癌细胞破坏，例如，在每次化疗或放射治疗后。然后我计划为这种多阶段破坏性治疗模型开发一些推断方法。可靠性分析：对连贯的系统寿命数据进行了广泛的分析，使用了“系统签名”的概念。已经开发出用于系统和使用签名的部件的寿命特征的有效的推论方法。但是，这些发展大多是关于两国制度的(无论是向上还是向下)。在这里，我计划考虑具有二值分量的多态相干系统，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我们的推理方法。分布理论：文献中已经研究了几种二元和多元离散分布。我计划使用COM-Poisson模型、一般的二元Poisson模型和复合来发展灵活的多元离散分布，并为它们开发推理方法。有序数据分析和随机排序：最近讨论了比例风险、比例反向风险和尺寸族中的最小值和最大值的各种随机排序，以及它们在精算中的应用。我计划将这些结果推广到一般的顺序统计量，并将其推广到相依情况。对该领域和加拿大的预期结果和好处：我指导的大多数博士后和博士后都在许多大学任职，包括加拿大的卡尔加里、马尼托巴省、滑铁卢、渥太华和麦克马斯特，以及美国的普渡大学、锡拉丘兹大学、南卫理公会大学、明尼苏达州、加拿大和得克萨斯州。还有一些人加入了加拿大公司，如庞巴迪、蒙特利尔银行、加拿大丰业银行、加拿大帝国商业银行、TD银行、GE Capital、罗杰斯和加拿大轮胎公司。我预计拟议的研究工作将产生类似的结果和好处。

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.