Sequential estimation of parameters in compound poisson processes

Abstract

This thesis is concerned with two sequential estimation procedures for compound Poisson processes. One is an efficient sequential estimation procedure and the other is a Bayes sequential estimation procedure. In efficient sequential estimation procedure, the cases where the jump sizes are exponential class random variables are considered. Cramer-Rao type information inequality gives the efficiency criterion. Unbiased estimators of which the variances attain the lower bound are all characterized with the corresponding sampling plans. In Bayes sequential estimation procedure, the case where the jump sizes are Bernoulli random variables is considered. When at most one sampling stage is available, optimal decision rule minimizing the Bayes risk, and the optimal stopping and continuation regions are obtained. The optimality of those regions for infinite horizon problem is discussed and the finiteness of the stopping time is also shown.