Empirical Likelihood Ratio Tests For Homogeneity Of Multiple Populations

Homogeneity testing of populations is one of the most important topics in the field of statis-tics,and has important application value in the fields of society,economy and medicine.When the population distributions are known,especially in the normal population case,there are many classical parameter test methods to solve the problem[5],such as,analysis of variance,T test,F test and chi-square test,likelihood ratio test,and some new methods about homogeneity testing of populations which are developed by many scholars based on above tests.While the population distributions are unknown,there are few relevant literatures to research the problem of homogene-ity testing of multiple populations.The Krusal-Wallis test(KWT)proposed by Kruskal and Wallis[1]is one famous test in this aspect,which is applied to nonparametric testing of two or more pop-ulations Owen[2.3]established the empirical likelihood(EL)method for nonparametric situation systematically.Since then,many studies have shown that EL has significant advantages compared to other commonly used statistical methods(such as approximation method etc.).So many re-searchers have applied this method to the various types of statistical inference,such as Albert el al.[43]have studied an empirical likelihood ratios based goodness-of-fit test based on the sample entropy;Safavinejad el al.[45]have studied a density-based empirical likelihood ratios goodness-of-fit test for Rayleigh distribution and power comparison under the independent samples.Qin and Lawless[38]have used the empirical likelihood method to generalized estimating equations model etc.Thus,to the empirical likelihood method to deal with the problem of testing for homogeneity of multiple populations is of great significance.In this paper,we use the empirical likelihood method to construct the empirical likelihood ratio test(ELRT)statistics for homogeneity of multiple populations,and show that the asymptotic distribution of the statistic is the chi-square distribution under some regularity conditions when the null hypothesis is true.Thus the rejection region of the test is given,and the main conclusions of this paper are verified by numerical simulations,and the efficiency of the proposed test ELRT and KWT is compared through numerical simulations.The main content of this article is divided into three chapters:The first chapter is introduction.In this chapter,we briefly introduce the background of the homogeneity testing of populations distributions,the general situation of the research background of the empirical likelihood method,the content and innovation points of this article studied,the research situation of Kruskal-Wallis test,research background and research progress of empirical likelihood method,and the research content and innovation points of this paper.The second chapter will construct empirical likelihood ratio test statistics to solve hypothesis test problems of homogeneity of multiple populations,and give some assumptions and main results of this paper,the simulation results,lemmas and main results;The third chapter is the main conclusion and the future review of the topic related in this paper.The innovation of this article is as follows:1.It is the first time that the empirical likelihood method is used to construct test statistics for homogeneity of multiple populations,to prove the asymptotic distribution of the test statistics,and to obtain the rejection region of the testing;2.The main conclusions are verified by numerical simulations,and we also compare powers with KWT.The conclusion is that the powers of ELRT and KWT have no big difference when sample size is bigger,but the power of the ELRT is better than KWT's when the sample size is small.