Quantile regression with cubic polynomial splines under shape constraints with applications

The study describes a method for quantile smoothing with cubic splines, considering monotonicity or convexity constraints. It is built upon an initial idea of \citeapos{N1}, who used quadratic splines, that are sufficient for zero order estimation, modal detection, or other estimators implying the first derivative. The use of cubic splines will not only improve convergence, but also allows the uniform convergence of the second derivatives, useful in many applications among which inflection point detection. On the other hand, while mean regression is based on $L_2$ norm, quantile regression uses asymmetric standard deviation (with $L_1$ type norm) and allows the exploration of the whole range of a distribution. We recall the convergence of the constrained estimator in case non-vanishing derivative, and give a proof in case of vanishing derivative, or mutiple consraints/variations. The method is applied to very current problems such as global warming or inflection point detection for Covid infection data. In the appendix, we give details about a proof of the uniform convergence of the estimator, although there exits general ones, and also compute the $B$-spline coefficients on the monomial basis, useful to the theoretical statement of our programming code.