Second-order Edgeworth Expansion For One-sample Studentized U-statistics

The theory of Edgeworth expansion dates back more than a cen-tery. The typical application of Edgeworth expansion is to approximate the true distributions of statistics. Many authors have studied the properties of the Edgeworth expansions for the distributions of Ustatistics since the invention. Because of the complexity of the terms in the expansions, many turned to the research of the second-order (also called one-term) Edgeworth expansions, and the rate of convergence of the distributions of the standardized and studentized U-statistics to the normality has been the central problem of all. Many results have been obtained about the second-order Edgeworth expansions for the studentized U-statistics. More recently, Edgeworth expansions are developed for a general class of statistics under rather weak conditions and applications of those results are given to obtain Edgeworth expansions for the distributions of the studentized U-statistics constructed from one independent and identically distributed (i.i.d.) sample under weaker moment conditions than those available in the literature before.However, due to the facts of complexity of the terms in the expansions, nearly all the results are too difficult to apply directly. In this thesis, we will give some detailed formulae of the coefficients in the expansions for the convenience of application when the sample is coming from a particular distribution and the kernel function is of some specific form. We will give some simulation results. In future work, we will try to show that the convergence rate remains unchanged under the same moment conditions when the U-statistics are constructed from two independent samples, i.e. o(n^-1/2...