Robust Estimation of Conditional Mean by the Linear Combination of Quantile Regressions

In this paper, we propose a new robust estimator for regression problems in the form of the linear combination of quantile regressions. The proposed robust regression estimator is helpful for the conditional mean estimation especially when the error distribution is asymmetric or/and heteroscedastic, where conventional robust regressions yield considerable bias to the conditional mean. First we introduce a wide class of heteroscedastic forms what we call generalized location scale models and show that the linear combination of quantile regressions can be a consistent conditional mean estimator under generalized location scale models. Next we formulate the proposed robust regression estimator and investigate its statistical properties by influence function. The investigation clarifies the robustness and an interesting property of the proposed estimator. Some numerical experiments and an application to insurance premium estimation problem are shown to illustrate the effectiveness of the proposed estimator. keywords: robust regression, quantile regression, conditional mean estimator, asymmetric error, heteroscedastic error, generalized location scale models, influence function, insurance premium estimation