Asymptotics of bivariate penalised splines. (English) Zbl 1420.62180

Author’s abstract: We study the class of bivariate penalised splines that use tensor product splines and a smoothness penalty. Similar to G. Claeskens et al. [Biometrika 96, No. 3, 529–544 (2009; Zbl 1170.62031)] for the univariate penalised splines, we show that, depending on the number of knots and penalty, the global asymptotic convergence rate of bivariate penalised splines is either similar to that of tensor product regression splines or to that of thin plate splines. In each scenario, the bivariate penalised splines are found rate optimal in the sense of C. J. Stone [Ann. Stat. 10, 1040–1053 (1982; Zbl 0511.62048)] for a corresponding class of functions with appropriate smoothness. For the scenario where a small number of knots is used, we obtain expressions for the local asymptotic bias and variance and derive the point-wise and uniform asymptotic normality. The theoretical results are applicable to tensor product regression splines.


62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
62G15 Nonparametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference


Full Text: DOI


[1] Barrow, D.; Smith, P., Efficient \(####\) Approximation by Splines, Numerische Mathematik, 33, 101-114 (1979) · Zbl 0442.41015
[2] Barrow, D. L.; Smith, P. W., Asymptotic Properties of Best \(####\)[0, 1] Approximation by Splines with Variable Knots, Quarterly of Applied Mathematics, 36, 3, 293-304 (1978) · Zbl 0395.41012
[3] Chen, H.; Wang, Y.; Paik, M. C.; Choi, H. A., A Marginal Approach to Reduced-Rank Penalized Spline Smoothing with Application to Multilevel Functional Data, Journal of the American Statistical Association, 108, 504, 1216-1229 (2013) · Zbl 1288.62155
[4] Claeskens, G.; Krivobokova, T.; Opsomer, J. D., Asymptotic Properties of Penalized Spline Estimators, Biometrika, 96, 3, 529-544 (2009) · Zbl 1170.62031
[5] De Boor, C. (1976), ‘Splines as Linear Combinations of B-splines’, in Approximation Theory, II, eds. G. Lorentz, C. Chu and L. Schumaker, New York: Academic Press, pp. 1-47. · Zbl 0343.41011
[6] De Boor, C., A Practical Guide to Splines (1978), Berlin: Springer, Berlin · Zbl 0406.41003
[7] Demko, S., Inverses of Band Matrices and Local Convergence of Spline Projections, SIAM Journal on Numerical Analysis, 14, 4, 616-619 (1977) · Zbl 0367.65024
[8] Eilers, P.; Marx, B., Multivariate Calibration with Temperature Interaction Using Two-Dimensional Penalized Signal Regression, Chemometrics and Intelligent Laboratory Systems, 66, 159-174 (2003)
[9] Eilers, P.; Marx, B.; Durbán, M., Twenty Years of P-splines, SORT, 39, 2, 1149-186 (2015)
[10] Hall, P.; Opsomer, J. D., Theory for Penalised Spline Regression, Biometrika, 92, 1, 105-118 (2005) · Zbl 1068.62045
[11] Huang, J. Z., Local Asymptotics for Polynomial Spline Regression, The Annals of Statistics, 31, 5, 1600-1635 (2003) · Zbl 1042.62035
[12] Kauermann, G.; Krivobokova, T.; Fahrmeir, L., Some Asymptotic Results on Generalized Penalized Spline Smoothing, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71, 2, 487-503 (2009) · Zbl 1248.62055
[13] Kiefer, J.; Wolfowitz, J., On the Deviations of the Empiric Distribution Function of Vector Chance Variables, Transactions of the American Mathematical Society, 87, 173-186 (1958) · Zbl 0088.11305
[14] Lai, M.-J.; Wang, L., Bivariate Penalized Splines for Regression, Statistica Sinica, 23, 3, 1399-1417 (2013) · Zbl 06202712
[15] Li, Y.; Ruppert, D., On the Asymptotics of Penalized Splines, Biometrika, 95, 2, 415-436 (2008) · Zbl 1437.62540
[16] Petrov, V., Sums of Independent Random Variables (1975), New York: Springer, New York · Zbl 0322.60042
[17] Ruppert, D.; Wand, M.; Carroll, R. J., Semiparametric Regression During 2003-2007, Electronic Journal of Statistics, 3, 1193-1256 (2009) · Zbl 1326.62094
[18] Schumaker, L., Spline Functions: Basic Theory (1981), Cambridge: Wiley-Interscience · Zbl 0449.41004
[19] Schwarz, K.; Krivobokova, T., A Unified Framework for Spline Estimators, Biometrika, 103, 1, 121-131 (2016) · Zbl 1452.62292
[20] Seber, G., A Matrix Handbook for Statisticians (2007), New Jersey: Wiley-Interscience, New Jersey
[21] Stone, C. J., Optimal Global Rates of Convergence for Nonparametric Regression, The Annals of Statistics, 10, 4, 1040-1053 (121982) · Zbl 0511.62048
[22] Wahba, G., Spline Models for Observational Data (1990), Philadelphia: Society for Industrial and Applied Mathematics · Zbl 0813.62001
[23] Wang, X.; Shen, J.; Ruppert, D., On the Asymptotics of Penalized Spline Smoothing, Electronic Journal of Statistics, 5, 1-17 (2011) · Zbl 1274.65012
[24] Wood, S., Thin Plate Regression Splines, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65, 95-114 (2003) · Zbl 1063.62059
[25] Wood, S., Low-rank Scale-invariant Tensor Product Smooths for Generalized Additive Mixed Models, Biometrics, 62, 1025-1036 (2006) · Zbl 1116.62076
[26] Xiao, L., Li, Y., Apanasovich, T., and Ruppert, D. (2012), Local asymptotics of P-splines. Unpublished technical report. Available at .
[27] Xiao, L.; Li, Y.; Ruppert, D., Fast Bivariate P-splines: The Sandwich Smoother, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75, 577-599 (2013) · Zbl 1411.62109
[28] Zhou, S.; Shen, X.; Wolfe, D. A., Local Asymptotics for Regression Splines and Confidence Regions, The Annals of Statistics, 26, 5, 1760-1782 (1998) · Zbl 0929.62052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.