Empirical Bayesian Estimation Of Parameters In Skew-normal Mixed Effects Model And Its Application

Mixed effect model is widely applied in the field of longitudinal data processing.In the existing research on this model,it's generally assumed that the random effect and error term obey the normal distribution.But the actual data are more commonly presented with non-symmetric,multi-peak and other non-normal distribution features.If we continue to use the traditional normal distribution assumption for statistical inference,it's easy to lack robustness and reduce statistical accuracy.On the other hand,most of the existing studies use least square estimation and maximum likelihood estimation to estimate the parameters of the model.However,these methods only focus on the current data,and fail to make full use of the prior distribution information of variables.In view of this,this paper use empirical Bayesian method to estimate the fixed and skewed parameters of the model,which combines with the prior information of data.Firstly,based on the conjugate distribution theory in the parameter prior distribution,the prior distribution of the fixed effect and the skewed parameter are determined.Then,the hyper-parameter and model parameters are divided into the interesting parameters and the redundant parameters.Secondly,according to the skew-normal density function of the model,the posterior density function of the interest parameters and the likelihood function of the redundant parameters are given by using Bayesian theorem in the form of random variables.Furthermore,the maximum likelihood estimation and MCMC techniques are combined to estimate the parameters.Then,the empirical Bayesian estimation of the skewed parameter and the fixed effect and its algorithm are obtained.Finally,the empirical Bayesian estimation is applied to the analysis of influencing factors of logistics demand in central part of China.The simulation result shows that in the case of small sample size,when changing the dimension and distribution of the fixed effect,the empirical Bayesian estimation is better than the Nelder-Mead estimation in sense of mean squared error.In the process of changing the distribution of skewed parameter,the empirical Bayesian estimation is still better than the Nelder-Mead estimation.The accuracy advantage for the estimation of skewed parameter is particularly obvious.In the case of large samples,empirical Bayesian estimation still has obvious advantages.However,as the sample size increases,the advantages of empirical Bayesian estimation over Nelder-Mead estimation are gradually reduced.In the actual case application,it's not only proved that the skew-normal mixed effect model has better fitness compared with the normal mixed effect model,but further verified the statistical superiority of empirical Bayesian estimation.