Mathematical Sciences: Research in Sequential Design, Optimal Stopping and Survival Analysis

本研究包括三个部分。第一部分涉及顺序实验的设计，其中寻求分配受试者的最优，渐近最优或近似最优规则。包括实际设计，其中外部和可能相反的约束影响可用规则的类别。主题包括多阶段规则，有偏见的硬币设计，重复显著性检验，强盗问题，以及利用历史控制的贝叶斯设计。第二部分涉及最优停止问题，主要侧重于推导接近最优的简单易用程序。本文还研究了一些广义秘书问题的最优解。第三部分涉及调整存活率估计值的方法，以考虑随机审查的影响。在寻找可接受的校正项时，将参数转换为具有良好行为的似然函数的技术与导出后验分布的可积渐近展开式的技术相结合。这项工作包括在一系列实验中对受试者进行最佳分配，以最大限度地获得信息。它包括对何时停止实验的问题的分析，通过确定已经获得了足够的信息来做出决定。这项工作还涉及对受试者存活率的估计，同时考虑到对某些受试者的观察结果是随机审查的事实。

This research contains three parts. The first part concerns the design of sequential experiments in which optimal, asymptotically optimal or approximately optimal rules for allocating subjects to treatments are sought. Included are practical designs, in which external and possibly, opposing constraints affect the class of rules available. Topics include multi-stage rules, biased coin designs, repeated significance tests, bandit problems, and Bayesian designs which utilize historical controls. The second part involves problems of optimal stopping with the main emphasis on derivations of simple-to-use procedures that are nearly optimal. Optimal solutions to some generalized secretary problems are also considered. The third part concerns methods of adjusting estimators of survival rates to account for effects of random censoring. In locating acceptable correction terms, techniques of transforming parameters to have well-behaved likelihood functions are combined with those of deriving integrable asymptotic expansions for posterior distributions. The work involves the study of the optimal assignment of subjects to treatment groups in a sequence of experiments to maximize the amount of information obtained. It includes the analysis of the question of when to stop the experiment by deciding that enough information has been obtained to reach a decision. The work also involves the estimation of survival rates of subjects while accounting for the fact that observations on some of the subjects are randomly censored.