Simultaneous confidence bands for extremal quantile regression with splines

Abstract

This study investigates simultaneous confidence bands for extremal quantile regressions using the spline method. We construct the spline estimator for intermediate order quantiles using a conventional quantile regression framework, and we obtain the extreme order quantile estimator by extrapolating the spline estimator for intermediate order quantiles. We establish the asymptotic normality of the spline and extrapolated estimators for intermediate and extreme order quantiles. By applying the volume of tube formula to the above two estimators, we construct simultaneous conditional quantile confidence bands for intermediate and extreme order quantiles. To confirm the performance of the proposed confidence bands, we use a Monte Carlo simulation and an example with real data.