Asymptotically Optimal Bayesian And Minimax Sequential Point Estimation.

Abstract

Four related problems are considered in this thesis. The first problem consists of determining an optimal stopping time for estimating the failure rate of an exponential distribution with a general, smooth loss function and a gamma prior, using censored data. The construction of such a stopping time involves obtaining asymptotic expansions for the posterior expected loss, computing the infinitesimal operator of some resulting Markov process, and using Dynkin's identity. The second problem is to obtain an asymptotic lower bound for the minimax regret of a sequential estimation procedure, in the case of a one-parameter exponential family, for a general class of estimators of the population mean and with a general, smooth loss function. The program is to determine the limit of the Bayes regret and then maximize with respect to the prior distribution. The third problem is to develop a uniform version of the non-linear renewal theorem. Finally, the fourth problem is to find a stopping time which attains the asymptotic lower bound of the second problem. The procedure used to yield such a stopping time requires the results developed in the third problem.