Bayesian ”confidence intervals” for the cross-validated smoothing spline. (English) Zbl 0538.65006

Wahba, Grace

J. R. Stat. Soc., Ser. B 45, 133-150 (1983).

The author considers the model \(Y(t_ i)=g(t_ i)+\epsilon_ i\), \(i=1,...,n,t_ i\in [0,1]\), where g(t) is a smooth function on [0,1] and \(\epsilon_ i\) are independent \(N(0,\tau^ 2)\)-errors with \(\tau^ 2\) unknowns. She studies ”confidence intervals” for the cross-validated smoothing spline estimate of g. The paper presents an affirmative answer (in the case of large n) to the question: Is there any reason to believe that the resulting 95 per cent confidence intervals will cover the true \(g(t_ i)\) about 95 per cent of the time?

Reviewer: B.D.Bojanov

MSC:

65D10	Numerical smoothing, curve fitting
65D07	Numerical computation using splines
41A15	Spline approximation
62F25	Parametric tolerance and confidence regions
62A01	Foundations and philosophical topics in statistics
65C99	Probabilistic methods, stochastic differential equations

Keywords:

Bayes estimates; spline smoothing; cross-validation; confidence intervals