Model Averaging For Bayesian Quantile Regression

With the development of quantile theory,Bayesian estimation of quantile regression has become an important direction of modern statistical research.The parameter estimation of Bayesian quantile regression(BQR)is usually based on a single model,which is the best model selected from a set of candidate models.However,this precess ignores the uncertainty of existing models and leads to overconfidence in inference and risk decisions in parameter estimation,and model averaging method can be a useful method to complement of BQR model selection.Focusing on the model averaging theory under the two frameworks of Bayesian and frequency,this paper applies the model averaging methods of S-AIC,S-BIC,Mallows and BMA to the BQR model.Furthermore,a model averaging method is proposed based on J-fold cross-validation,which selects data-driven weights by minimizing the value of quantile loss in J-fold cross-validation.Meanwhile,theorem 3.1 is given and proved about the asymptotic optimality for the model averaging estimator of the obtained BQR.In order to test the performance of the proposed model weight selection method under limited samples,numerical simulation experiments were carried out with other four model averaging methods under different sample sizes.Experimental results show that the model averaging method of BQR proposed is more balanced than other methods in model weight distribution,and has superior generalization ability.Finally,in order to further verify the effectiveness of the proposed model weight selection method,the proposed method is compared with the other four methods under the Immunoglobulin-G and wage datasets.The empirical results show that although the training error of the new method is not minimal though,but generalization ability is better than other models.After calculation,the prediction error of the model average method is much lower than the prediction error of the optimal single model,so the generalization ability of the model averaging method is much higher than that of a single model.