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On the asymptotics of penalized spline smoothing. (English) Zbl 1274.65012

Summary: 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.

MSC:

65C60 Computational problems in statistics (MSC2010)
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation

Software:

SemiPar; gss
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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. · Zbl 0954.62045 · doi:10.2307/3318440
[2] Claeskens, G., Krivobokova, T. and Opsomer, J. (2009). Asymptotic properties of penalized spline estimators., Biometrika , 96 , 529-544. · Zbl 1170.62031 · doi:10.1093/biomet/asp035
[3] de Boor, C. (2001)., A Practical Guide to Splines . Springer. · Zbl 0987.65015
[4] Eggermont, P. P. B. and LaRicci, V. N. (2006a). Equivalent kernels for smoothing splines., Journal of Integral Equations and Applications , 18 , 197-225. · Zbl 1139.34025 · doi:10.1216/jiea/1181075379
[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. · Zbl 1117.62039 · doi:10.1214/074921706000000879
[6] Eggermont, P. P. B. and LaRicci, V. N. (2009)., Maximum Penalized Likelihood estimation . Volume II: Regression. New York: Springer. · Zbl 1184.62063 · doi:10.1007/b12285
[7] Eiler, P. and Marx, B. (1996). Flexible smoothing with B-splines and penalties (with comments and rejoinder)., Statistical Science , 11 , 89-121. · Zbl 0955.62562 · doi:10.1214/ss/1038425655
[8] Eubank, R. L. (1999)., Nonparametric Regression and Spline Smoothing . New York: Marcek Dekker. · Zbl 0936.62044
[9] Eubank, R. L. (1999)., A Kalman Filter Primer , CRC Press, Boca Raton. · Zbl 1277.62017
[10] Gu, C. (2002)., Smoothing Spline: ANOVA Models . New York: Springer. · Zbl 1051.62034
[11] Hall, P. and Opsomer, J.D. (2005). Theory for penalised spline regression., Biometrika , 92 , 105-118. · Zbl 1068.62045 · doi:10.1093/biomet/92.1.105
[12] Koml oś, J., Major, P. and Tusn a\' dy, G. (1975). An approximation of partial sums of independent r.v.’s and the sample d.f., Z. Wahrsch. Verw. Gebiete , 32 , 111-131. · Zbl 0308.60029 · doi:10.1007/BF00533093
[13] Li, Y. and Ruppert, D. (2008). On the asymptotics of penalized splines., Biometrika , 95 , 415-436. · Zbl 1437.62540 · doi:10.1093/biomet/asn010
[14] Marx, B. and Eilers, P. (1996). Flexible smoothing with B-splines and penalties (with comments and rejoinder)., Statistical Science , 11 , 89-121. · Zbl 0955.62562 · doi:10.1214/ss/1038425655
[15] Marx, B. and Eilers, P. (2005). Multidimensional penalized signal regression., Technometrics , 47 , 13-22. · doi:10.1198/004017004000000626
[16] Messer, K. (1991). A comparison of a spline estimate to its equivelent kernel estimate., Annals of Statistics , 19 , 817-829. · Zbl 0741.62040 · doi:10.1214/aos/1176348122
[17] Messer, K. and Goldstein, L. (1993). A new class of kernels for nonparametric curve estimation., Annals of Statistics , 21 , 179-196. · Zbl 0777.62041 · doi:10.1214/aos/1176349021
[18] Nychka, D. (1995). Splines as local smoothers., Annals of Statistics , 23 , 1175-1197. · Zbl 0842.62025 · doi:10.1214/aos/1176324704
[19] O’Sullivan, F. (1986). A statistical perspective on ill-posed inverse problems (with Discussion), Statistical Science , 1 , 505-527. · Zbl 0625.62110 · doi:10.1214/ss/1177013525
[20] Rice, J. and Rosenblatt, M. (1983). Smoothing splines: regression, derivatives and deconvolution., Annals of Statistics , 11 , 141-156. · Zbl 0535.41019 · doi:10.1214/aos/1176346065
[21] Ruppert, D. (2002). Selecting the number of knots for penalized splines., Journal of Computational and Graphical Statisitcs , 11 , 735-757. · doi:10.1198/106186002853
[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. · Zbl 0547.62024 · doi:10.1214/aos/1176346710
[24] Stone, C.J. (1982). Optimal rate of convergence for nonparametric regression., Annals of Statistics , 10 , 1040-1053. · Zbl 0511.62048 · doi:10.1214/aos/1176345969
[25] Wahba, G. (1990), Spline Models for Observational Data . Philadelphia, PA: SIAM. · Zbl 0813.62001
[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. · Zbl 1146.62030 · doi:10.1111/j.1467-842X.2008.00507.x
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