The Asympototic Property Of The Weighted Multivariate Empirical Processes And The Local Times Of Two-Parameter Brownian Bridge

Statistical inference theory is always based on the results of random sam-ple X1,X2,…,Xn of the population X. They are independent and identically distributed random elements with the same distribution as the population X. Generally, X is k-dimensional, and its distribution function is unknown. How to infer reasonably for the unknown distribution F is the most basic research subjects in the theory of non-parametric statistical inference.Usually,let and call Fn as the empirical distribution of random sample andβnF the em-pirical process of random sample. The empirical process theory established by Glivenko-Cantelli theorem,Kolmogorov-Smirnov statistical theory,Donsker theo-rem etc. shows that the empirical distribution function and the empirical process are the most valid tool to solve it directly.The convergency of empirical processes are turned to be the core of the problem when the goodness-of-fit between the empirical distribution function and the true distribution function is considered, Furthermore, by the Skorohod theorem and Donsker theorems, it is well known that Brownian bridge is the limiting process of the empirical processes in some senses. With the development of research and its application,weighted empirical processes are gradually concerned by a lot of scholars over the last two decade. and research in this area have emerged from time to time.However, the existed results are more complete for the empirical distribution function and the empirical process in the case of k=1. When k≥2,that is,the research on the multivariate empirical process and multi-parameter Brownian bridge have many areas to be perfect. The paper attempts to do some researches on the local times of two-parameter Brownian bridge and a weighted multivariate empirical process. The local times of two-parameter Brownian bridge and the asymptotic properties of the weighted empirical processes are obtained.The organisation of this paper is as follows:Chapter 1 begins with some background of the research, and the investiga-tive situation and main results of related issues at home and abroad are included.In Chapter 2,some notations and basic concepts are introduced, and mean-while some basic knowledge is raised in the derivation of this article.In Chapter 3,the elementary properties of multi-parameter Brownian bridge are considered.Chapter 4 aims to present a general result of convergence of the weighted empirical process by discussing the convergence of the weighted uniform empirical process.In Chapter 5,we discusse the local times of the multivariate empirical process and multi-parameter Brownian bridge.Finally, we obtain that the local time of the multivariate empirical process will converge to the one of multi-parameter Brownian bridge.