Electronic Journal of Statistics

On the asymptotics of penalized spline smoothing

Xiao Wang, Jinglai Shen, and David Ruppert

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Abstract

This paper performs an asymptotic analysis of penalized spline estimators. We compare P-splines and splines with a penalty of the type used with smoothing splines. The asymptotic rates of the supremum norm of the difference between these two estimators over compact subsets of the interior and over the entire interval are established. It is shown that a P-spline and a smoothing spline are asymptotically equivalent provided that the number of knots of the P-spline is large enough, and the two estimators have the same equivalent kernels for both interior points and boundary points.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 1-17.

Dates
First available in Project Euclid: 14 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1295040628

Digital Object Identifier
doi:10.1214/10-EJS593

Mathematical Reviews number (MathSciNet)
MR2763795

Zentralblatt MATH identifier
1274.65012

Subjects
Primary: 62G08: Nonparametric regression 62G20: Asymptotic properties
Secondary: 62G05: Estimation

Keywords
Boundary kernel difference penalty equivalent kernel Green’s function P-spline

Citation

Wang, Xiao; Shen, Jinglai; Ruppert, David. On the asymptotics of penalized spline smoothing. Electron. J. Statist. 5 (2011), 1--17. doi:10.1214/10-EJS593. https://projecteuclid.org/euclid.ejs/1295040628


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References

  • [1] Abramovich, F. and Grinshtein, V. (1999). Deriviation of equivalent kernel for general spline smoothing: a systematic approach., Bernoulli, 5, 359–379.
  • [2] Claeskens, G., Krivobokova, T. and Opsomer, J. (2009). Asymptotic properties of penalized spline estimators., Biometrika, 96, 529–544.
  • [3] de Boor, C. (2001)., A Practical Guide to Splines. Springer.
  • [4] Eggermont, P. P. B. and LaRicci, V. N. (2006a). Equivalent kernels for smoothing splines., Journal of Integral Equations and Applications, 18, 197–225.
  • [5] Eggermont, P. P. B. and LaRicci, V. N. (2006b). Uniform error bounds for smoothing splines., IMS Lecture Notes—Monograph Series High Dimensional Probability 51, 220–237.
  • [6] Eggermont, P. P. B. and LaRicci, V. N. (2009)., Maximum Penalized Likelihood estimation. Volume II: Regression. New York: Springer.
  • [7] Eiler, P. and Marx, B. (1996). Flexible smoothing with B-splines and penalties (with comments and rejoinder)., Statistical Science, 11, 89–121.
  • [8] Eubank, R. L. (1999)., Nonparametric Regression and Spline Smoothing. New York: Marcek Dekker.
  • [9] Eubank, R. L. (1999)., A Kalman Filter Primer, CRC Press, Boca Raton.
  • [10] Gu, C. (2002)., Smoothing Spline: ANOVA Models. New York: Springer.
  • [11] Hall, P. and Opsomer, J.D. (2005). Theory for penalised spline regression., Biometrika, 92, 105–118.
  • [12], Komls, J., Major, P. and Tusndy, G. (1975). An approximation of partial sums of independent r.v.’s and the sample d.f., Z. Wahrsch. Verw. Gebiete, 32, 111–131.
  • [13] Li, Y. and Ruppert, D. (2008). On the asymptotics of penalized splines., Biometrika, 95, 415–436.
  • [14] Marx, B. and Eilers, P. (1996). Flexible smoothing with B-splines and penalties (with comments and rejoinder)., Statistical Science, 11, 89–121.
  • [15] Marx, B. and Eilers, P. (2005). Multidimensional penalized signal regression., Technometrics, 47, 13–22.
  • [16] Messer, K. (1991). A comparison of a spline estimate to its equivelent kernel estimate., Annals of Statistics, 19, 817–829.
  • [17] Messer, K. and Goldstein, L. (1993). A new class of kernels for nonparametric curve estimation., Annals of Statistics, 21, 179–196.
  • [18] Nychka, D. (1995). Splines as local smoothers., Annals of Statistics, 23, 1175–1197.
  • [19] O’Sullivan, F. (1986). A statistical perspective on ill-posed inverse problems (with Discussion), Statistical Science, 1, 505–527.
  • [20] Rice, J. and Rosenblatt, M. (1983). Smoothing splines: regression, derivatives and deconvolution., Annals of Statistics, 11, 141–156.
  • [21] Ruppert, D. (2002). Selecting the number of knots for penalized splines., Journal of Computational and Graphical Statisitcs, 11, 735–757.
  • [22] Ruppert, D., Wand, M.P., and Carroll, R.J. (2003)., Semiparametric Regression. Cambridge: Cambridge University Press.
  • [23] Silverman, B.W. (1984). Spline smoothing: the equivalent variable kernel method., Annals of Statistics, 12, 898–916.
  • [24] Stone, C.J. (1982). Optimal rate of convergence for nonparametric regression., Annals of Statistics, 10, 1040–1053.
  • [25] Wahba, G. (1990), Spline Models for Observational Data. Philadelphia, PA: SIAM.
  • [26] Wand, M.P. and Ormerod, J.T. (2008) On semiparametric regression with O’Sullivan penalized splines., Austral. New Zeal. J. Statist., 50, 179–198.