Parameter Estimation And Application Of Finite Multivariate Skew-normal Mixture Models

Since K.Pearson applied the two-component normal mixing model to deal with Crab data in 1894[63],a lot of research has been done on the problem of finite mixed normal model,which has become an important theory in the field of statistics and solved many problems in practice.But in real life,the data studied often show the characteristics of asymmetry and heavy tail,which makes the traditional assumption of normality of data face great challenges.If we assume the normality of data with asymmetry and heavy tail,we will get wrong statistical inference and poor fitting ac-curacy.However,data with this kind of structure are more common in real lifeIn order to deal with this kind of data,Azzalini proposed the concept of monadic skewed normal distribution in 1986[11],which can more effectively deal with one-dimensional data with skew and asymmetry,and the normal distribution is a special case of skewed normal distribution.In view of the excellent characteristics of the finite mixed distribution,many scholars tried to replace the component density function of the mixed distribution with the skewed normal density function after 2000,which greatly improved the fitting accuracy of the data and reduced the error of parameter estimation.However,because the density function of the skewed normal distribution con-tains the density function of the skewed normal distribution,the finite mixture model with the component of the skewed normal distribution also has the shortcoming of the classical finite normal mixture model-the problem of the likelihood function being unbounded.Secondly,although the value of skewness parameter will not cause the likelihood function to be unbounded,when the skewness parameter is too large,it will lead to increased calculation burden and bad statistical inferenceIn this paper,we add a penalty term to the scale parameter and skewness pa-rameter in the likelihood for several types of multivariate finite mixed skewed normal distribution models to offset the influence of the unbounded likelihood caused by the s-cale parameter being too small and the skewness parameter being too large.The theory and error of parameter estimation based on EM algorithm and the consistency theory of parameters are mainly studied.Smaller estimation errors are obtained in numerical simulation and actual data analysisThe research content of this thesis is mainly divided into five chaptersIn the first chapter,we introduce the research background and review the basic theory of finite mixture model and some of its basic propertiesIn the second chapter,we mainly study the consistency theory and parameter estimation of monadic mixed skewed normal distribution,and the low order moment theory of this distributionIn the third chapter,we mainly study the consistency theory and parameter es-timation of multivariate mixed skewed normal distribution,and study the low-order moment theory of this distribution.The 2-d and 3-d conditions of 2 components and 3 components were numerically simulated,and the AIS data were analyzedIn chapter 4,we mainly study the parameter estimation problem of multivariate mixed skewed normal distribution based on the definition of ellipsoid distribution,and study the low-order moment theory of this distributionIn chapter 5,we mainly study the parameter estimation problem of multivariate mixed skewness distribution with scale independent distribution density and the calcu-lation formula of low order moment.