• Bernoulli
  • Volume 5, Number 2 (1999), 359-379.

Derivation of equivalent kernel for general spline smoothing: a systematic approach

Felix Abramovich and Vadim Grinshtein

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We consider first the spline smoothing nonparametric estimation with variable smoothing parameter and arbitrary design density function and show that the corresponding equivalent kernel can be approximated by the Green function of a certain linear differential operator. Furthermore, we propose to use the standard (in applied mathematics and engineering) method for asymptotic solution of linear differential equations, known as the Wentzel-Kramers-Brillouin method, for systematic derivation of an asymptotically equivalent kernel in this general case. The corresponding results for polynomial splines are a special case of the general solution. Then, we show how these ideas can be directly extended to the very general L-spline smoothing.

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Bernoulli, Volume 5, Number 2 (1999), 359-379.

First available in Project Euclid: 5 March 2007

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Green's function L-smoothing spline


Abramovich, Felix; Grinshtein, Vadim. Derivation of equivalent kernel for general spline smoothing: a systematic approach. Bernoulli 5 (1999), no. 2, 359--379.

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  • [1] Abramovich, F. (1993) The asymptotic mean squared error of L-smoothing splines. Statist. Probab. Lett., 18, 179-182.
  • [2] Abramovich, F. and Steinberg, D. M. (1996) Improved inference in nonparametric regression using Lk - smoothing splines. J. Statist. Plann. Infer., 49, 327-341.
  • [3] Ansley, C. F., Kohn, R. and Wong, C.-H. (1993) Nonparametric spline regression with prior information. Biometrika, 80, 75-88.
  • [4] Bender, C. M. and Orszag, S. A. (1978) Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill.
  • [5] Coddington, E. A. and Levinson, N. (1955) Theory of Ordinary Differential Equations. New York: McGraw-Hill.
  • [6] Cox, D. D. (1983) Asymptotics for M-type smoothing spline. Ann. Statist., 11, 530-551.
  • [7] Eubank, R. L. (1988) Spline Smoothing and Nonparametric Regression. New York: Dekker.
  • [8] Gu, C. and Wahba, G. (1993) Smoothing spline ANOVA with component-wise Bayesian ''confidence intervals''. J. Comput. Graph. Statist., 2, 97-117.
  • [9] Hutchinson, M. and de Hoog, F. (1986) Smoothing data with spline functions. Numer. Math., 47, 99-106.
  • [10] Jeffreys, H. (1924) On certain approximate solutions of linear differential equations of the second order. London Math. Soc., 23, 428-436.
  • [11] Kimeldorf, G. S. and Wahba, G. (1971) Some results on Tchebycheffian spline functions. J. Math. Anal. Applic., 33, 82-95.
  • [12] Kohn, R. and Ansley, C. F. (1983) On the smoothness properties of the best linear unbiased estimate of a stochastic process observed with noise. Ann. Statist., 11, 1011-1017.
  • [13] Kohn, R. and Ansley, C. F. (1988) Equivalence between Bayesian smoothness priors and optimal smoothing for function estimation. In J. C. Spall (ed.), Bayesian Analysis of Time Series and Dynamic Models, pp. 393-430. New York: Dekker.
  • [14] Messer, K. (1991) A comparison of a spline estimate to its equivalent kernel estimate. Ann. Statist., 19, 817-829.
  • [15] Messer, K. and Goldstein, L. (1993) A new class of kernels for nonparametric curve estimation. Ann. Statist., 21, 179-195.
  • [16] Nussbaum, M. (1985) Spline smoothing in regression and asymptotic efficiency in L2. Ann. Statist., 13, 984-997.
  • [17] Nychka, D. W. (1988) Bayesian confidence intervals for a smoothing spline. J. Amer. Statist. Assoc., 83, 1134-1143.
  • [18] Nychka, G. W. (1995) Splines as local smoothers. Ann. Statist., 23, 1175-1197.
  • [19] Oehlert, G. W. (1992) Relaxed boundary smoothing splines. Ann. Statist., 20, 146-160.
  • [20] Priestley, M. B. and Chao, M. T. (1972) Non-parametric function fitting. J. Roy. Statist. Soc. Ser. B, 34, 385-392.
  • [21] Silverman, B. W. (1984) Spline smoothing: the equivalent variable kernel method. Ann. Statist., 12, 898-916.
  • [22] Silverman, B. W. (1985) Some aspects of the spline smoothing approach to non-parametric regression curve fitting. J. Roy. Statist. Soc., Ser. B, 47, 1-52 (with discussion).
  • [23] Smirnov, V. I. (1964) A Course of Higher Mathematics, Vol. 5. Oxford: Pergamon.
  • [24] Speckman, P. (1981) The asymptotic integrated mean square error for smoothing noisy data by splines. University of Oregon Preprint.
  • [25] Speckman, P. (1985) Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist., 13, 970-983.
  • [26] Tapia, R. A. and Thompson, J. R. (1978) Nonparametric Probability Density Estimation. Baltimore, MD: Johns Hopkins University Press.
  • [27] Wahba, G. (1978) Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. Roy. Statist. Soc. Ser. B, 40, 364-372.
  • [28] Wahba, G. (1983) Bayesian 'confidence intervals' for the cross-validated smoothing spline. J. Roy. Statist. Soc. Ser. B, 45, 133-150.
  • [29] Wahba, G. (1985) A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann. Statist., 13, 1378-1402.
  • [30] Wahba, G. (1990) Spline Models for Observational Data. Philadelphia, PA: SIAM.
  • [31] Wecker, W. E. and Ansley, C. F. (1983) The signal extraction approach to nonlinear regression and spline smoothing. J. Amer. Statist. Assoc., 78, 81-89.