Research On The Number Of Components In Finite Mixture Models Based On The Modified Likelihood Ratio Test

The test for the number of components in finite mixture models is one of main research topics in model selection.If this problem is well solved,it will have a profound impact on biology,genetics,medicine,economics,engineering and other fields.A modified likelihood ratio test enjoys an elegant asymptotic theory in parameter testing.However,several problems commonly exist in classical methods,such as over-estimation of the number of components,breakdown of regularity conditions and complicated asymptotic null distribution of the likelihood ratio test.The modified likelihood ratio test method can avoid the above defects.Yet there is still not a complete solution for asymptotic theory in model selection.This research is to supplement asymptotic theory.A negative-definiteness condition is introduced and the asymptotic null distribution is obtained.In this work,a negative-definiteness condition is introduced to gamma mixture model and normal mixture model.An asymptotic result for the null distribution and the asymptotic null distribution in different mixture models are presented.Mixture models with constraint on parameters are also discussed.The theoretical results prove that the asymptotic theory still holds for mixture models with constraint on parameters and without constraint on parameters.Then extensive simulations are conducted to verify the accuracy of the theory.In addition,a detailed analysis of parameter value selection is carried out according to the simulation results,including the penalty term value,the initial value of parameter and the number of iteration.Whether the parameter value has influence on precision will be verified by every single factor analysis.The simulation results show the theoretical results are consistent with simulated results.Real data examples involving house price data from different cities in China and gene expression data are included.Through the application of a modified likelihood ratio of hypothesis test method,to choose the more suitable mathematical model and calculate the corresponding parameter values,likelihood ratio values and number of mixture components.For the house price data analysis,according to the latest evaluation of tier,mathematical model for each tier is obtained.For gene expression data analysis,ten areas of brain are studied,the differentially expressed is separated from the non-differentially expressed transcripts.Therefore,this work not only supplements computational verification of the asymptotic theory for the modified likelihood ratio test and generalizing the mixture models in the application of gene expression,but also provides an exemplary method for solving the asymptotic distribution.