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Spline models for observational data. (English) Zbl 0813.62001
CBMS-NSF Regional Conference Series in Applied Mathematics. 59. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. XII, 169 p. (1990).
Statisticians are generally interested in smoothing data of the form \[ y_ i= f(x_ i)+ \varepsilon_ i, \qquad i=1,2,\dots, n, \] where \(\varepsilon_ i\) are random disturbances and \(f\) is only known to be “smooth”. A formal statement of the problem may be formulated as follows: find \(f\) in a given class of smooth functions \(W\) on the interval \((a,b)\) to minimize (for some \(\lambda>0\)) \[ {1\over n} \sum_{i=1}^ n (y_ i- f(x_ i))^ 2+ \lambda \int_ a^ b (f^{(m)} (x))^ 2 dx. \] All splines considered in the book are solutions to variational problems. The variational problems are treated from a unified point of view as optimization problems in reproducing kernel Hilbert spaces and it is assumed that the reader has a knowledge of the basic properties of Hilbert spaces.
Contents: 1. Background (1.1. Positive-definite functions, covariances, and reproducing kernels; 1.2. Reproducing kernel spaces on \([0,1]\) with norms involving derivatives; 1.3. The special and general spline smoothing problems; 1.4. The duality between reproducing kernel Hilbert spaces and stochastic processes; 1.5. The smoothing spline and the generalized smoothing spline as Bayes estimates).
2. More splines (2.1. Splines on the circle; 2.2. Splines on the sphere, the role of the iterated Laplacian; 2.3. Vector splines on the sphere; 2.4. The thin-plate spline on \(E^ d\); 2.5. Another look at the Bayes model behind the thin-plate spline).
3. Equivalence and perpendicularity, or What’s So Special About Splines? (3.1. Equivalence and perpendicularity of probability measures; 3.2. Implications for kriging).
4. Estimating the smoothing parameter (4.1. The importance of a good choice of \(\lambda\); 4.2. Ordinary cross validation and the “leaving- out-one” lemma; 4.3. Generalized cross validation (GCV); 4.4. Properties of the GCV estimate of \(\lambda\); 4.5. Convergence rates with the optimal \(\lambda\); 4.6. Other estimates of \(\lambda\) similar to GCV; 4.7. More on other estimates; 4.8. The generalized maximum likelihood estimate of \(\lambda\); 4.9. Limits of GCV).
5. Confidence intervals (5.1. Bayesian confidence intervals; 5.2. Estimate-based bootstrapping).
6. Partial spline models (6.1. Estimation; 6.2. Convergence of partial spline estimates; 6.3 Testing).
7. Finite-dimensional approximating subspaces (7.1. Quadrature formulae, computing with basis functions; 7.2. Regression splines).
8. Fredholm integral equations of the first kind (8.1. Existence of solutions, the method of regularization; 8.2. Further remarks on ill- posedness; 8.3. Mildly nonlinear integral equations; 8.4. The optimal \(\lambda\) for loss functions other than predictive mean-square error).
9. Further nonlinear generalizations (9.1. Partial spline models in nonlinear regression; 9.2. Penalized GLIM models; 9.3. Estimation of the log-likelihood ratio; 9.4. Linear inequality constraints; 9.5. Inequality constraints in ill-posed problems; 9.6. Constrained nonlinear optimization with basis functions; 9.7. System identification).
10. Additive and interaction splines (10.1. Variational problems with multiple smoothing parameters; 10.2. Additive and interaction smoothing splines).
11. Numerical methods.
12. Special topics (12.1. The notion of “high frequency” in different spaces; 12.2. Optimal quadrature and experimental design).
The bibliography contains more than three hundred items. The book is based on a series of 10 lectures at Ohio State University at Columbus in March 23-27, 1987.

62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G07 Density estimation
65C99 Probabilistic methods, stochastic differential equations
65D07 Numerical computation using splines
65R20 Numerical methods for integral equations
62G20 Asymptotic properties of nonparametric inference
45B05 Fredholm integral equations