Likelihood Tests And Goodness Of Fit Tests Under Associated Samples

This thesis investigates the hypothesis testing problem of multivariate parameter space under associated samples. Under some mild conditions, the asymptotic distribu-tions of the likelihood ratio statistic and goodness of fit statistic are obtained. This thesis has three chapters:The first chapter is introduction. We introduce research situation of associated sample, research progress of likelihood ratio statistic, research survey of goodness of fit statistic, the research content and innovation points.The second chapter studies the asymptotic distributions of the likelihood ratio statistic under associated samples. We utilize the form of likelihood function under independent samples, considering the specific forms of the likelihood ratio statistic in-cluding simple and composite null hypotheses are given. The asymptotic distributions under two situation are proved to be x2-distribution and the simulation results show our results perform well.The third chapter studies the asymptotic distributions of the goodness of fit statistic under associated samples. We build the testing statistic in two cases on the basis of the overall sample’s parameter, then we find that the asymptotic distributions under two situation are x2-distribution after simplificating the rank of matrix in virtue of the special properties of idempotent matrix.The innovation of this thesis is mainly reflected in the following three aspects:l.We generalize the conclusion to multidimensional parameter space from the the-orem under one-dimensional parameter space by dint of the multi-dimensional vector’s Taylor expansion and resolution of symmetric non negative definite matrix.2.We conduct simulation study that shows that our theoretical results perform well.3.The asymptotic distributions of the goodness of fit statistic are obtained by means of the special properties of idempotent matrix.