Fitting Mixed Erlang Densities Under Roughness Penalty

In statistics,the problem of how to give the density function of given data well by mathematical expression has drawn too much attention.Especially when it comes to the censored data or multi-nmode data,the density curve needs not only be of high flexibility Tbut,also can get rid of overfit,which greatly increase the difficulty of fitting.Among them,the development of mixed model has received people's attention,because the hybrid model is a semi-parametric model instead parametric model or non parametric model.It avoids the deviation problem between fitting curve and the raw data structure and provides mathematical expression of the fitting curve directly.After Tijims[2]proving that,in the sense of weak convergence,mixed Erlang model with a common scale parameter can converge to any positive distributions,which means mixed Erlang can fit any positive distribution with arbitrary precisions,the performance of mixed Erlang models is widely used in modeling financial and insurance data.The expression of mixed Erlang distributions is,(?)(?)where ? represents the weight of each Erlang branch in the mixed distribution,say a K weight vector,and satisfies 0<?k? 1,A:? 1E...,K,and(?)a commonscale parameter used by all Erlang dist.ribution branches.m=(m1,…,mK)is a set of shape pa.rameters,each component of which is a positive integer.K is the order of t,he mixed nmodel,that,is,the number of Erlang distribution branches.In Lee&Lin's[3,4],EM algorithm was introduced into the estimation of the mixed Erlang model.In fact,EM algorit.hm is a kind of iterative algorithms.Because of the strong dependence on the initial values,different initials may influence the fitting results a lot.In[7,8],the initial values of scale parameter came from a very large alternative space,through continuous iteration,the scale parameter with poor performance will be deleted.However,due to a large range of parameters,especially the order of the mixed model,it is prone to overfit.In this paper,CMM[7]initial method is used to determine the initial values of EM algorithm.The linear structure of the hybrid Erlang model is very good at heterogeneity,but it has an inevitable problem:the determination of the mixed number.Many scholars have discussed the determination of the mixtures of Gaussian models,which includes the minimum distance method,the hypothesis test method,the penalized likelihood method and so on.Fan&Li[10]proposed SCAD penalty on linear regression model to realize variable selection and regression coefficient determination.In Yin&Lin's paper[9],similarly to SCAD penalty function,the author proposed a new threshold penalty function,iSCAD penalty to select the order of the model,and the estimators also satisfy sparsity,continuity and unbiased properties.But in practice,the penalty algorithm converge slowly.Based on the above,this paper proposed another penalized likelihood function,the penalty function here is inspired by the traditional roughness penalty.It's three squared derivative of the continuous density function.The roughness penalty function is defined as follows:(?)(?)We illustrate the performance of the proposed method by some simulations and real data studies in chapter 4,and the results reveal a better performance compared with non-penalized method generally.