Approximations for moments and distributions in sequential analysis

The stopping rules in sequential methods have posed a lot of difficulties in analyzing the efficiencies of sequential procedures. The exact distributions of the stopping points and the statistics related to those stopping rules are hardly available explicitly. Woodroofe (1976), (1977), Lai and Siegmund (1977), (1979) and Aras and Woodroofe (1993) have laid a foundation for making asymptotic analysis for a large class of stopping rules. In this thesis we consider the moments and distributions of some randomly stopped standardized summations. Refined moment expansions are derived after simplifying a result of Zhang (1988) in nonlinear renewal theory, and moreover, Edgeworth expansion type of approximations for the distributions are provided by means of Fourier transformations. A rigorous mathematical study for a subclass cases while Mykland (1993)'s martingale expansion is applicable shows that the expansions from these two different ways are the same. Their applications in sequential estimations and estimations after sequential tests are established. The second order coverage of a confidence interval by Chow-Robbins procedure, the bias and mean squared error of the maximum likelihood estimator after a sequential test as well as the skewness of its distribution are presented. Simulation studies are conducted for both sequential estimation and estimation after sequential test, which show that the approximations we obtained work very well