## The average posterior variance of a smoothing spline and a consistent estimate of the average squared error.(English)Zbl 0731.62084

Summary: A smoothing spline estimator can be interpreted in two ways: either as the solution to a variational problem or as the posterior mean when a particular Gaussian prior is placed on the unknown regression function. In order to explain the remarkable performance of her Bayesian “confidence intervals” in a simulation study, G. Wahba [J. R. Stat. Soc., Ser. B 40, 364-372 (1978; Zbl 0407.62048)] conjectured that the average posterior variance of a spline estimate evaluated at the observation points will be close to the expected average squared error.
The estimate of the average posterior variance proposed by Wahba is shown to converge in probability to a quantity proportional to the expected average squared error. This result is established by relating this statistic to a consistent risk estimate based on generalized cross- validation.

### MSC:

 62G07 Density estimation 62G15 Nonparametric tolerance and confidence regions

Zbl 0407.62048
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