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Penalized spline regression is a popular method for scatterplot smoothing,but there has long been a debate on how to construct confidence intervals forpenalized spline fits. Due to the penalty, the fitted smooth curve is a biasedestimate of the target function. Many methods, including Bayesian intervals andthe simple-shift bias-reduction, have been proposed to upgrade the coverage ofthe confidence intervals, but these methods usually fail to adequately improvethe situation at predictor values where the function is sharply curved. In thispaper, we develop a novel approach to improving the confidence intervals byusing a smaller smoothing strength than that of the spline fits. With acarefully selected amount of reduction in smoothing strength, the confidenceintervals achieve nearly nominal coverage without being excessively wide orwiggly. The coverage performance of the proposed method is investigated viasimulation experiments in comparison with the bias-correction techniquesproposed by Hodges (2013) and Kuusela and Panaretos (2015).

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