Quantile regression with monotonicity restrictions using P-splines and the L1-norm

Abstract

Quantile regression is an alternative to OLS regression. In quantile regression, the sum of absolute deviations or the L1-norm is minimized, whereas the sum of squared deviations or the L2-norm is minimized in OLS regression. Quantile regression has the advantage over OLS-regression of being more robust to outlying observations. Furthermore, quantile regression provides information complementing the information provided by OLS-regression. In this study, a non-parametric approach to quantile regression is presented, which constrains the estimated-quantile function to be monotone increasing. In particular, P-splines with an additional asymmetric penalty enforcing monotonicity are used within an L1-framework. This can be translated into a linear programming problem, which will be solved using an interior point algorithm. As an illustration, the presented approach will be applied to estimate quantile growth curves and quantile antibody levels as a function of age.