Theory and Applications of U-statistics for Multistate Models under Censoring

U-统计量是统计学理论和方法的基本对象。它们是样本均值的推广，直接或间接地涵盖了估计量、估计函数（对于参数的每个值）和检验统计量的整个范围。这个建议的目的是系统地调查一类这些统计量，可能会出现在各种多态删失数据问题下的一个统一的框架。多状态模型是渐进系统，其中个体从一个状态移动到下一个状态，并且所产生的数据是事件时间数据的概括。感兴趣的变量包括各种转换时间、进入和离开给定状态的时间、给定瞬态中的等待时间等，或者所有上述和附加协变量的混合（如果有的话）。考虑右删失和区间删失多态数据。本文还研究了U-统计量理论和方法在多状态非参数检验和置信区间构造中的应用。 多态数据出现在不同的领域和应用中。例如，这些状态可能对应于患者的健康状况或住房市场的强度或世界各地的气候条件等。因此，预计由此产生的方法将为包括医学、市场营销、政治学和工程学在内的多个学科的研究人员提供有价值的统计工具。拟议的研究还将大大有助于培训未来的统计人员，因为其中部分内容将在PI的监督下用于博士论文，部分研究结果将纳入研究生课程。

U-statistics are fundamental objects in the theory and methods of statistics. They are generalizations of sample means and cover directly or indirectly a whole range of estimators, estimating functions (for each value of the parameter) and test statistics. The purpose of this proposal is to investigate systematically a class of these statistics that may arise in various multistate censored data problems under a unified framework. Multistate models are progressive systems where individuals move from one state to the next and the resulting data are generalizations of event time data. The variables of interest include various transition times, entry and exist times to and from a given state, waiting times in a given transient state and so on or a mixture of all of the above and additional covariates, if any. Both right and interval censored multistate data are considered. Applications of the U-statistics theory and methods to the construction of nonparametric tests and confidence intervals in multistate problems are also studied. Multistate data arise in diverse fields and applications. For example, the states may correspond to the health status of patients or the strength of the housing market or the climatic conditions in various parts of the world and so on. As a result, it is expected that the resulting methods will provide valuable statistical tools to researchers in multiple disciplines including medicine, marketing, political science and engineering. The proposed research will also contribute substantially in training future statisticians since parts of it will be used for doctoral dissertations under PI's supervision and some of the findings will be incorporated in graduate level courses.