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

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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.