Inference for penalized spline regression: Improving confidence intervals by reducing the penalty

Abstract: Penalized spline regression is a popular method for scatterplot smoothing, but there has long been a debate on how to construct confidence intervals for penalized spline fits. Due to the penalty, the fitted smooth curve is a biased estimate of the target function. Many methods, including Bayesian intervals and the simple-shift bias-reduction, have been proposed to upgrade the coverage of the confidence intervals, but these methods usually fail to adequately improve the situation at predictor values where the function is sharply curved. In this paper, we develop a novel approach to improving the confidence intervals by using a smaller smoothing strength than that of the spline fits. With a carefully selected amount of reduction in smoothing strength, the confidence intervals achieve nearly nominal coverage without being excessively wide or wiggly. The coverage performance of the proposed method is investigated via simulation experiments in comparison with the bias-correction techniques proposed by Hodges (2013) and Kuusela and Panaretos (2015).