Confidence bands for smoothness in nonparametric regression

The choice of the smoothing parameter in nonparametric regression is critical to the form of the estimated curve and any inference that follows. Many methods are available that will generate a single choice for this parameter. Here we argue that the considerable uncertainty in this choice should be explicitly represented. The construction of standard simultaneous confidence bands in nonparametric regression often requires difficult mathematical arguments. We question their practical utility, presenting several deficiencies. We propose a new kind of confidence band that reflects the uncertainty regarding the smoothness of the estimate.