Simultaneous Confidence Bands for Penalized Spline

In this paper we construct simultaneous confidence bands for a smooth curve using penalized spline estimators. We consider three types of estimation methods: (i) as a standard (fixed effect) nonparametric model, (ii) using the mixed model framework with the spline coefficients as random effects and (iii) a full Bayesian approach. The volume-of-tube formula is applied for the first two methods and compared from a frequentist perspective to Bayesian simultaneous confidence bands. It is shown that the mixed model formulation of penalized splines can help to obtain, at least approximately, confidence bands with either Bayesian or frequentist proper-ties. Simulations and data analysis support the methods proposed. The R package ConfBands accompanies the paper. Key words and phrases. Bayesian penalized splines; B-splines; Confidence band; Mixed model; Penalization