Adaptive semiparametric M-quantile regression

Parametric and semiparametric regression beyond the mean have become important tools for multivariate data analysis in this world of heteroscedasticity. Among several alternatives, quantile regression is a very popular choice if regression on more than a location measure is desired. This is also due to the inherent robustness of a quantile estimate. However, when moving towards the tails of a distribution, the handling of extreme observations becomes crucial for empirical estimates. M-quantiles handle outliers within the regression analysis by imposing a strong robustness to the loss function. However, this loss function is typically not designed to handle heteroscedasticity. An adaptive extension to the degree of robustness within the loss function is proposed along with the implementation of semiparametric predictors in an M-quantile regression model. A practical method to compute confidence intervals is also presented. The methods are supported by extensive simulations and an analysis of childhood malnutrition in Tanzania.