Bayes Regression Based On Asymmetric Quadratic Loss Function

When it comes to the estimation of regression coefficients, the most basic method is the least squares method. However, the least squares method has a lot of shortcomings. On one hand, it neglects the influence of covariates on the end of the response variables. In some practical problems, the tail is usually the focus of our attention. On the other hand, the assumption of random error term is too stringent. Economic data generally exhibit tail spikes characteristics, so the least squares estimation will no longer have the statistical superiority.1978, Koenker and Bassett first proposed the concept of quantile regression, the concept is a popularization of median regression. Compared with ordinary least squares method, quintile regression model not only do not need to make any assumptions on the overall distribution, but also describe the independent variable X for the dependent variable Y range accurately, catch the tail characteristics of the distribution. With the development of computer theory, Bayesian inference method has been widely used in many areas. Bayesian estimation method is superior to the conventional method. Combined with Bayesian statistical inference methods and quantile regression, we get the Bayesian quantile regression. Therefore, Bayesian quantile regression not only obtains comprehensive description of sample information, but also improves the accuracy of forecasts.In this paper, Bayesian estimation based on asymmetric squared loss function was proposed. It is an extention of the least squares estimation and Bayesian quantile regression estimation. In Bayesian framework, we find the equivalent maximum likelihood estimation, and.get the conclusion that if we choose the prior of β to be improper uniform,then the resulting joint posterior distribution will be proper. Finally, through simulation,we prove that in some cases the Bayesian estimation based on asymmetric squared loss function is better than least squares estimation and Bayesian quantile regression estimation.