Bayesian Confidence Intervals for Smoothing Splines

Abstract

The frequency properties of Wahba's Bayesian confidence intervals for smoothing splines are investigated by a large-sample approximation and by a simulation study. When the coverage probabilities for these pointwise confidence intervals are averaged across the observation points, the average coverage probability (ACP) should be close to the nominal level. From a frequency point of view, this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error. These properties are independent of the Bayesian assumptions used to derive this confidence procedure, and they explain why the ACP is accurate for functions that are much smoother than the sample paths prescribed by the prior. This analysis accounts for the choice of the smoothing parameter (bandwidth) using cross-validation. In the case of natural splines an adaptive method for avoiding boundary effects is considered. The main disadvantage of this approach is that these confidence intervals are only valid in an average sense and may not be reliable if only evaluated at peaks or troughs in the estimate.

Key Words:

• Boundary effects
• Cross-validation
• Nonparametric regression
• Smoothing parameter