Bayesian “Confidence Intervals” for the Cross-Validated Smoothing Spline

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Abstract
We consider the model Y(ti)=g(ti) + ∈i, i = 1, 2, …, n, where g(t), t ∈ [0, 1] is a smooth function and the {∈i} are independent N(0, σ2) errors with σ2 unknown. The cross‐validated smoothing spline can be used to estimate g non‐parametrically from observations on Y(ti), i= 1, 2, …, n, and the purpose of this paper is to study confidence intervals for this estimate. Properties of smoothing splines as Bayes estimates are used to derive confidence intervals based on the posterior covariance function of the estimate. A small Monte Carlo study with the cubic smoothing spline is carried out to suggest by example to what extent the resulting 95 per cent confidence intervals can be expected to cover about 95 per cent of the true (but in practice unknown) values of g(ti), i= 1,2,…,n. The method was also applied to one example of a two‐dimensional thin plate smoothing spline. An asymptotic theoretical argument is presented to explain why the method can be expected to work on fixed smooth functions (like those tried), which are “smoother” than the sample functions from the prior distributions on which the confidence interval theory is based.