Studies On Some Topics In Finite Mixture Models

Finite mixture models possess much flexibility and convenience in analyzing data which may include two or more sub-populations.Consequently,finite mixture models are applied in many areas,for instance,astronomy,medicine,genetics,engineering,social science and so on.The backgrounds of this thesis are from medicine and genetics,and they are closely related to mixture models.The main contents are following three parts.In the first part,we study the homogeneity in a two-sample problem in which one of the sample maybe from two component mixture model.For this problem,we assume the kernel function from a general location-scale distribution family,and we do not require two scale parameters of kernel functions equal.The appearance of mixture distribution and the inequality of scale parameters lead to that the likelihood function is unbounded and the Fisher information about mixing proportion maybe infinite.These undesirable properties bring a great challenge for the two-sample problem.To overcome the challenge,the penalized likelihood function is established and based on it,the EM testing statistic is proposed.The EM testing statistic can detect the mean information and variance information simultaneously.In addition,we investigate the limiting distribution of the EM testing statistic under the null and local alternative.The inference of sample size is also discussed.Simulation results and real data examples show that the proposed EM test are more powerful than the existing methods and it is applicable in practice.The contribution of this part extends the application of EM test in two-sample problem under the general location-scale mixture model.Quantitative trait locus(QTL)interval mapping always involves mixture models.In the second part,assuming the kernel function from location-scale distribution families,we study the likelihood ratio test(LRT)in the QTL interval mapping problems,which contains two genetic situations:in the meiosis,double recombination between the non-sister chromatids of one homologous chromosome does not occur and it occurs.The research work corresponding the two situations are respectively provided in the third chapter and the fourth chapter.In the third chapter,we presents the large-sample properties of the maximum likeli-hood estimators of the unknown parameters and the likelihood ratio tests(LRTs)under two situations:(1)the kernel functions may have different locations and/or scales,and(2)the kernel functions have the same unknown scale.Under(1)and(2),the two limit-ing distributions are respectively the supreme of ?22(?)and the supreme of X12(?).Under these results,determining critical values of the LRTs is still very difficult.To handle the problem,we further derive the explicit representations of the two limiting distributions.Based on the explicit representations,the critical values are easily calculated.In addition,we also investigate the limiting distributions of likelihood ratio statistic under the local alternative.The finite-sample performance of the LRTs and comparison with the existing methods are investigated via simulation studies.Simulation results indirectly show the excellent properties we derive for LRTs.In the final of this chapter,a real data example is analyzed,which shows that the LRTs are applicable.In the fourth chapter,we assume the double recombination between the non-sister chromatids of one homologous chromosome occurs,other assumptions are the same with those in the third chapter.For the presentation of double recombination,the statistical model is no longer same with that in the third chapter.We also consider the situations(1)and(2)mentioned in the previous paragraph.Under(2),the construction of the LRT and its asymptotic properties are similar to those of the third chapter.It should be noted that under(1),we could not construct the LRT directly based on the likelihood function.That is because under the statistical model,the likelihood function is unbounded,which leads that the consistent maximum likelihood estimators do not exist.Consequently,we add penalty functions on scale parameters,and we can get the consistent penalized maximum likelihood estimators through maximizing the penalized likelihood function.Based on the penalized likelihood function,the LRT is established and its asymptotic properties are investigated.Similar to the third chapter,the explicit representations and the limiting distributions under local alternatives are also investigated.The LRTs are compared with other existing methods via simulation and real data analysis.In the third part,for the finite location-scale mixture model with a structural param-eter,we study the strong consistency of the maximum likelihood estimators.Without any restriction on parameter space,the strong consistency conclusion and its rigorous proof are provided.In addition,some examples are listed out:finite normal mixtures,logistic mixtures,extreme value mixtures and t mixtures,and we show that these models satisfy our assumptions.