Robust Estimation Of A Two-component Semiparametric Mixture Model

Since the finite semiparametric mixture model was proposed,it has increasingly attracted attention in theoretical research and application due to its extremely flexible model hypothesis.Indeed,practical problems exsist in many fields,such as finance,economics,social sciences and biomedicine,in which we often encounter heterogeneous population data.So the two-component mixture model becomes an indispensable statistical analysis tool in heterogeniety.In this paper,we will focus on studying a class of two-component semi-parametric mixture models which are highly frequently quoted.After studying some existing parameter estimation methods,we proposed a new robust estimation method that was based on the continuous scale mixture.The new estimation method assumes that the unknown distributions in the model come from the rich mixed distribution classes of continuous normal scale.According to Lindsay's(1983a)conclusion,we demonstrated the existence and uniqueness of non-parametric maximum likelihood estimation for unknown density measures,and put forward our new method in combination with CNM algorithm proposed by Wang(2007).In addition,this paper introduced four estimation methods including Minimum profile hellinger distance estimation(MPHD)proposed by Xiang et al.(2014),Symmetrization put forward by Bordes et al.(2006),and EM-type and Maximizing ?type presented by Song et al.(2010).And in the simulation experiment and empirical analysis,our method was compared with these four methods.The results showed that our method could provide robust estimation in the case of unknown component model,unknown model belonging to mixed distribution classes,unknown model heavy tail,unknown model non-normal distribution or data containing outliers.The main contributions of this paper can be illustrated from the following three aspects:1.In a class of two-component mixture model studied in this paper,we assumed that one of the unknown distribution comes from the ? distribution family,which contains all the symmetrical probability density distributions with unimodal peaks.Even if the true component density does not belong to the distribution family ?,our estimation program can work well.Based on this assumption,we also demonstrated the identifiability of the model.Moreover,with this new modeling method,we can properly avoid the problem of misdesignation in the traditional parametric mixture model.2.This paper proposed an estimation method based on continuous scale mixture on the basis of non-parametric estimation CNM algorithm.Therefore,this paper briefly described the basic theory and implementation steps of the algorithm,and proved the incremental nature of the algorithm,which strengthened the theoretical basis for our proposed method.3.Different from some existing semi-parametric estimation methods,the proposed method does not need to modify or smooth the likelihood.Besides,in simulation and empirical analysis,we proved that the proposed estimation method had advantages and it could provide robust estimators while the component model was unknown or the data contained outliers.