Parameter Estimation And Application Of Generalized Pareto Mixtures Via SCAD Penalty

In this paper,we consider the estimation of a two parameters generalized Pareto mixture model(Hereinafter referred to as generalized Pareto mixture model)with density:where p =(p1,…,pm),0<Pj<1 is a mixed weight coefficient with ?j=1m pj=1.and ?=(?1,…,Am),?=(?1,…,?m),?j,?j>0,j=1,…,m.The mixture model fits the real data very well.Due to the characteristics of thick tail,the Pareto distribution has been paid more and more attention in the field of financial risk measurement.At the same time,it has been widely used in insurance and reliability analysis.In practice,we often need to use the generalized Pareto mixture model.In this paper,we select the most widely used class of generalized Pareto distribution.The traditional method of moment estimation and maximum likelihood estimation was used to estimate the parameters of mixture distribution.It is found that the traditional method of moment estimation and maximum likelihood estimation which is very good.EM algorithm is an iterative algorithm to estimate the maximum likelihood which is very common.It can be carried out on the maximum likelihood estimation of parameters from incomplete data,calculated the posterior density function.The biggest advantage of the proposed algorithm is simple and stable,but easy to fall into local optimum,selection is very dependent on the initial value.K-means clustering provides a method for selecting initial value of iteration.However,how to cluster the number of mixed distribution of the number of branches m becomes a problem.In this paper,we introduce the SCAD penalty to remove the redundant branch density function by punishing the weight parameter,so as to solve the problem that the number of branches is unknown and the initial value of iteration is chosen.Finally,we use K-S two-independent sample homogeneity test model.There are three contributions in this paper:(1)The SCAD penalty is introduced to prove that the SCAD penalized likelihood estimator for the generalized Pareto mixture model is uniform.(2)Calculated the iterative formula for parameter estimation of the generalized Pareto mixture model,the introduction of SCAD punishment,and overcome the problem of how to determine the mixing order m,finally through the MATLAB and R software numerical simulation to verify the feasibility of the improved EM algorithm.(3)The distribution of population density of China's provinces and autonomous regions in 2016 and the Hurun Report of China rich list in the year of 2016 by via the generalized Pareto mixture model.It is proved that EM algorithm via the SCAD penalty can be used to estimate the generalized Pareto mixture model in the case of various sample sizes,and the results are very good.