Bayesian Variable Selection And Estimation Based On Asymmetric Squared Loss

Least squares estimation no longer has excellent properties when there is heteroskedasticity in the data.Quantile regression shows excellent property under heteroscedasticity,but due to its absolute value loss,the computational efficiency of quantile regression is always one of the primary considerations.Asymmetric least squares regression not only deals with heteroscedasticity,but also overcomes the difficulties of quantile regression.The heteroscedasticity mainly comes from the measurement error and the influence of some factors omitted in the model on the response variable,and the phenomenon of high-dimensional heteroscedastic data is more common.Therefore,it is particularly important to find a method suitable for processing high-dimensional heteroscedastic regression.This thesis considers the problems of variable selection and estimation under heteroscedasticity from a Bayesian perspective.Compared with the methods of frequentist,the advantage of the Bayesian method is that it takes into account the prior distribution of the parameters,so that it can solve practical problems quickly and accurately.Therefore,it is important to study the variable selection and estimation of linear regression model under heteroscedasticity from the Bayesian perspective.Based on a prior function suitable for high-dimensional data,which is called Spike-and-Slab Lasso prior,we firstly proves the large sample properties of the prior under asymmetric square loss.It is found that the method has excellent theoretical properties,such as the effective posterior dimension and optimal posterior concentration.Secondly,we study the superiority of the method by simulation.In order to show the effectiveness of the proposed method in dealing with high-dimensional heteroscedastic data,we consider error homogeneity and two heteroscedastic structures.At the same time,three dimension cases of the explanatory variables are considered,which are p < n,p = n,p > n.We use MCMC algorithm for sampling.And we use the datasets generated from specific models to compare the proposed method with Bayesian Lasso,Bayesian adaptive Lasso,and Spike-and-Slab prior with point-mass mixture,the results show that the method proposed in this thesis has certain advantages in the accuracy of variable selection and estimation,and at different asymmetric levels τ,this method can successfully identify the heteroscedasticity of the data.Besides,the method proposed in this thesis is applied to the housing market data and gene(e QTL)data,and it is found that the prediction error of the method proposed in this thesis is smaller.The above results show that the method proposed in this thesis can be effectively used for high-dimensional heterogeneity data analysis.