Approximations for moments and distributions in sequential analysis

The stopping rules in sequential methods have posed a lot of difficulties inanalyzing the efficiencies of sequential procedures. The exact distributionsof the stopping points and the statistics related to those stopping rules arehardly available explicitly. Woodroofe (1976), (1977), Lai and Siegmund(1977), (1979) and Aras and Woodroofe (1993) have laid a foundation formaking asymptotic analysis for a large class of stopping rules.In this thesis we consider the moments and distributions of some randomlystopped standardized summations. Refined moment expansions are derivedafter simplifying a result of Zhang (1988) in nonlinear renewal theory, andmoreover, Edgeworth expansion type of approximations for the distributionsare provided by means of Fourier transformations. A rigorous mathematicalstudy for a subclass cases while Mykland (1993)'s martingale expansion isapplicable shows that the expansions from these two different ways are thesame. Their applications in sequential estimations and estimations aftersequential tests are established. The second order coverage of a confidenceinterval by Chow-Robbins procedure, the bias and mean squared error of themaximum likelihood estimator after a sequential test as well as the skewnessof its distribution are presented. Simulation studies are conducted for bothsequential estimation and estimation after sequential test, which show thatthe approximations we obtained work very well.