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On generalized cross-validation for multivariate smoothing spline functions. (English) Zbl 0622.65008
The aim of this paper is to contribute to the study of generalized cross- validation showing that it satisfied an asymptotic optimality condition and to prove that under the assumption that the knots have an asymptotic behavior defined by a cumulative distribution function with bounded density. To prove the main theorem (Th. 1.1), some basic inequalities are developed in section 2. In section 3, the eigenvalues of \({\mathfrak A}_ n(\lambda)\) and the behavior of tr(\({\mathfrak A}_ n)\lambda))\) and tr(\({\mathfrak A}^ 2_ n(\lambda))\) are studied, and then the proof of Th. 1.1 is given and the consistency results for generalized cross-validation (Th. 3.5) are obtained. The extension of these results to the case of thin splines over bounded domains is discussed in the last section.
Reviewer: Xie Shenquan

65D10 Numerical smoothing, curve fitting
65D07 Numerical computation using splines
41A15 Spline approximation
41A63 Multidimensional problems
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