Statistic Analysis Of Finite Mixture Models

Finite mixture models have generated strong and sustained interest for the last few decades, due to their usefulness as an extremely flexible method of modeling a wide variety of random phenomena. The fields of application in which mixture models have been found relevant have been extremely diverse throughout science, medicine, engineering and even humanities. However, because of the breakdown of the regularity conditions under standard theory, the asymptotic optimality of the likelihood ratio test is questionable and its distributional properties are also difficult to evaluate. As a result, analysis of mixture distributions is still a challenge for application of numerical methods in statistics and the test for homogeneity in the mixture models is difficult to study.The finite mixture models have density In this paper, we discuss the EM algorithm of exponential family mixture models, the power of the likelihood ratio test, the asymptotic distribution of the statistics and the linear models with explanation variables.In Chapter 2, we give general form of the EM algorithm and introduce the methods to solve the question of MLE. The EM algorithm is the standard tool for maximum likelihood estimation in finite mixture models. In particular, we derive the iterative formula of the EM algorithm for exponential family mixture models.In Chapter 3, we calculate the percentile points and simulate the power of four likelihood ratio test by the EM algorithm, which are based on different implementations of the likelihood maximization algorithm. We also show that global maximization of the likelihood is not appropriate to obtain a good power of the likelihood ratio test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, resulted in a rather powerful test.In Chapter 4, we discuss the asymptotic distribution of the likelihood ratio test statistics. A class of mixture model is discussed, where 0≤ξ≤1, f is probability density function. The asymptotic distribution of likelihood ratio statistic of the model is given.In Chapter 5, we apply the exponential family mixture models to solve the question of linear models and get some good results. The residual and influence analysis of the models are discussed carefully in this section.