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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 $$y\sb i= f(x\sb i)+ \varepsilon\sb i, \qquad i=1,2,\dots, n,$$ where $\varepsilon\sb i$ 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 $\lambda>0$) $${1\over n} \sum\sb{i=1}\sp n (y\sb i- f(x\sb i))\sp 2+ \lambda \int\sb a\sp b (f\sp{(m)} (x))\sp 2 dx.$$ 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 $E\sp d$; 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.

62-02Research monographs (statistics)
62G07Density estimation
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
65D07Splines (numerical methods)
65R20Integral equations (numerical methods)
62G20Nonparametric asymptotic efficiency
45B05Fredholm integral equations