Splines from a Bayesian point of view

Angers J. F. and Delampady, M. (1992). Hierarchical Bayesian curve, fitting and smoothing.Canadian J. Statist. 20, 35–49.

Ansley, C. F., Kohn, R. and Wong, C.-M. (1993). Nonparametric spline regression with prior information.Biometrika 80, 75–88.

Berger, J. O. (1988) (2nd. ed.).Statistical Decision Theory and Bayesian Analysis. Berlin: Springer.

Buckley, M. J., Eagleson, G. K. and Silverman, B. W. (1988). The estimation of residual variance in nonparametric regression.Biometrika 75, 189–199.

Cressie, N. (1991).Statistics for Spatial Data. New York: Wiley.

Diaconis, P. (1988). Bayesian numerical analysis.Statistical Decision Theory and Related Topics IV (S. S. Gupta, J. O. Berger, eds.), Berlin: Springer, 162–175.

Eubank, R. L. (1984) Approximate regression models and splines.Commun. Statist.-Theor. Meth. 13, 433–484.

Eubank, R. L. (1985) Diagnostics for smoothing splinesJ. Roy. Statist. Soc. B47, 332–341.

Eubank, R. L. (1988).Spline Smoothing and Nonparametric Regression. New York/Basel: Dekker

Green, P.J. and Silverman, B. (1994).Nonparametric Regression and Generalized Linear Models. London: Chapman and Hall.

Gu, Chong (1992). Penalized likelihood regression: A Bayesian analysis.Statistica Sinica 2, 255–264.

Hutchinson, M. F., Booth, T. H., Mc Mahon, J. P. and Nix, H. A. (1984). Estimating monthly mean values of daily total solar, radiation for Australia.Solar Energy 32, 277–290.

Jaynes, E. (1984). The intuitive inadequacy of classical statistics.Epistemiologia VII (Special Issue), 43–74.

Kimeldorf, G. S. and Wahba, G. (1970a). Spline functions and stochastic processes.Sankya A 132, 173–180.

Kimeldorf, G. S. and Wahba, G. (1970b). A correspondence between Bayesian estimation on stochastic processes and smoothing by splines.Ann. Math. Statist. 41, 495–502.

Kimeldorf, G. S. and Wahba, G. (1971). Some results on Tchebycheffian spline functions.J. Math. Anal. Appl. 33, 82–95.

Kohn, R. and Ansley, C. F. (1988) Equivalence between Bayesian smoothness priors and optimal smoothing for function estimation.Bayesian Analysis of Time Series and Dynamic Models (Spall, J. C., ed.), New York: Dekker, 393–430.

Kuelbs, J., Larkin, F. M. and Williamson, J. A. (1972). Weak probability distributions on reproducing kernel Hilbert spaces.Rocky Mt. J. Math. 2, 369–378.

Kuo, H. (1975).Gaussian measures in Banach spaces, Berlin, New York: Springer.

Larkin, F. M. (1972). Gaussian measure in Hilbert space and applications in numerical analysis.Rocky Mt. J. Math. 2, 379–421.

Larkin, F. M. (1980). A probabilistic approach to the estimation of functionals.Approximation Theory III (Cheney., E. W. ed.), London: Academic Press, 577–582.

Larkin, F. M. (1983). The weak Gaussian distribution as a means of localization in Hilbert space.Applied Nonlinear Functionalanalysis (Gorenflo, R. and Hoffmann, K.-H., eds.) Frankfurt a.M.: Verlag Peter Lang, 145–177.

Li, K.-C. (1982) Minimaxity of the Method of Regularization on Stochastic Processes.Ann. Satist. 10, 937–942.

van der Linde, A., Witzko, K.-H. and Jöckel, K.-H. (1995). Spatial-temporal analyses of mortality using splinesBiometrics, (to appear).

van der Linde, A. (1994). The invariance of statistical analyses with smoothing splines with respect to the inner product in the reproducing kernel Hilbert space.Compstat 94. (to appear).

van der Linde, A. (1993a). A note on smoothing splines as Bayesian estimates.Statistics and Decisions 11, 61–67.

van der Linde, A. (1993b). Smoothing splines with linear constraints.Proc. of the Interregional Meeting of the German and Netherlands Region of the International Biometric Society. Muenster, Germany, March 15–18, 1994.

Lindley, D. V. and Smith, A. F. M. (1972). Bayes estimates for the linear model.J. Roy. Statist. Soc. B 34, 1–41.

Meinguet, J. (1979). Multivariate interpolation at arbitrary points made simple.J. Appl. Math. Phys. (ZAMP) 30, 292–304.

Ripley, B. D. (1988).Statistical Inference for Spatial Processes, Cambridge: University Press.

Silverman, B. W. (1985). Some aspects of the spline smoothing approach to nonparametric regression curve fitting.J. Roy. Statist. Soc. B 47, 1–52, (with discussion).

Thomas-Agnan, C. (1990). A family of splines for non-parametric regression and their relationship with Kriging.Statistics 21, 533–548.

Thomas-Agnan, C. (1991). Spline functions and stochastic filtering.Ann. Statist. 19, 1512–1527.

Wahba, G. (1978). Improper priors, spline smoothing and the problem of guarding against model errors in regressions.J. Roy. Statist. Soc. B 40, 364–372.

Wahba, G. (1981) Nurnerical experiments with the thin plate histospline..Comm. Statist.-Theor. Meth. A 10, 2475–2514.

Wahba, G. (1983). Bayesian ‘Confidence Intervals’ for the cross-validated smoothing spline.J. Roy. Statist. Soc. B,45, 133–150.

Wahba, G. (1984) Cross-validated spline methods for the estimation of multivariate functions from data on functionals.Statistics: An Appraisal, (David, H.A. and David, H. T. eds.), Iowa State: Univ. Press, 205–235.

Wahba, G. (1990) Spline models for observational data, Philadelphia, Pennsylvania: SIAM.

Wahba, G. and Wendelberger, J. (1980). Some new mathematical methods for variational objective analysis using splines and cross-validation.Monthly, Weather Review 108, 1122–1145.

Wecker, W.E. and Ansley, C. F. (1983). The signal extraction approach to nonlinear regression and spline smoothing.J. Amer. Statist. Soc. 78, 81–89.

Weinert, H. L. (1978). Statistical methods in optimal curve fitting.Comm. Statist.-Simul. Comput. 7, 417–435.

Weinert, H.L. and Sidhu, G.S. (1978). A Stochastic Framework for Recursive Computation of Spline Functions, Part I, Interpolating Splines.IEEE Transactions on Information Theory,IT-24, 45–50.