Generalized Likelihood Ratio Test For Heteroscedastic Normal Mixtures

This paper mainly does some research about the test problem of the heteroscedastic normal mixtures.It uses the generalized maximum likelihood estimation(GMLE)and improves generalized likelihood ratio test(GLRT)so that it can apply to heteroscedastic normal mixtures.Under the normal location mixture model,we can use directly the R-package RE-Bayes to compute the GLRT because of the same Variance.So we need the lower bound of sample variance(?0)and use the equation of convolution to transfer the location-scale mixture model to the location model in the heteroscedastic normal mixtures.In the third chapter it verifies the equivalence between the normal location mixture model and the heteroscedastic normal mixtures.Thus the computing problem of heteroscedastic GLRT is solved.The fourth chapter conducts numerical simulation.Before the simulation,we need to get the lower bound of sample variance.However,because of the uncertainty of sample variance,we can't get a reliable lower bound.We find out that the misspecification of lower bound has no effect on the power of GLRT by simulation.Therefore in order to get the suitable lower bound ?0,it provides reference database for selecting ?0.In the paper,it does four kinds of numerical simulation to show the power of GLRT: the first is the sparse case;the second is the very sparse case;the third is the dense case;the forth is the random case.Under four kinds of cases,we compare the power of GLRT with others.We finally conclude that the power of GLRT is better and more robust.