×

zbMATH — the first resource for mathematics

Asymptotic optimality of a multivariate version of the generalized cross validation in adaptive smoothing splines. (English) Zbl 1282.62101
Summary: We consider an adaptive smoothing spline with a piecewise-constant penalty function \(\lambda(x)\), in which a univariate smoothing parameter \(\lambda\) in the classic smoothing spline is converted into an adaptive multivariate parameter \(\boldsymbol{\lambda}\). Choosing the optimal value of \(\boldsymbol{\lambda}\) is critical for obtaining desirable estimates. We propose to choose \(\boldsymbol{\lambda}\) by minimizing a multivariate version of the generalized cross validation function; the resulting estimator is shown to be consistent and asymptotically optimal under some general conditions, i.e., the counterparts of the nice asymptotic properties of the generalized cross validation in the ordinary smoothing spline are still provable. This provides a theoretical justification of adopting the multivariate version of the generalized cross validation principle in adaptive smoothing splines.
MSC:
62G08 Nonparametric regression and quantile regression
62H12 Estimation in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Abramovich, F. and Steinberg, D. M. (1996). Improved inference in nonparametric regression using \(L_k\)-smoothing splines., Journal of Statistical Planning and Inference 49 327-341. · Zbl 0881.62043
[2] Craven, P. and Wahba, G. (1979). Smoothing noisy data with spline functions., Numerische Mathematik 31 377-403. · Zbl 0377.65007
[3] Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage., Journal of the American Statistical Association 90 1200-1224. · Zbl 0869.62024
[4] Eubank, R. L. (1999)., Nonparametric Regression and Spline Smoothing . Marcel Dekker, New York. · Zbl 0936.62044
[5] Girard, D. A. (1991). Asymptotic optimality of the fast randomized versions of GCV and \(C_L\) in ridge regression and regularization., The Annals of Statistics 19 1950-1963. · Zbl 0754.62030
[6] Green, P. J. and Silverman, B. W. (1994)., Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach . Monographs on Statistics and Applied Probability 58 . Chapman & Hall, New York, NY. · Zbl 0832.62032
[7] Horn, R. and Johnson, C. R. (1985)., Matrix Analysis . Cambridge University Press, Cambridge. · Zbl 0576.15001
[8] Kim, H. and Huo, X. (2012). Locally optimal adaptive smoothing splines., Journal of Nonparametric Statistics 24 665-680. · Zbl 1254.62052
[9] Li, K. C. (1985). From Stein’s unbised risk estimates to the method of generalized cross validation., The Annals of Statistics 13 1352-1377. · Zbl 0605.62047
[10] Li, K. C. (1986). Asymptotic optimality of \(C_L\) and generalized cross-validation in ridge regression with application to spline smoothing., The Annals of Statistics 14 1101-1112. · Zbl 0629.62043
[11] Liu, Z. and Guo, W. (2010). Data driven adaptive spline smoothing., Statistica Sinica 20 1143-1163. · Zbl 05769960
[12] Mallows, C. L. (1973). Some comments on \(C_p\)., Technometrics 15 661-675. · Zbl 0269.62061
[13] Pintore, A., Speckman, P. and Holmes, C. C. (2006). Spatially adaptive smoothing splines., Biometrika 93 113-125. · Zbl 1152.62331
[14] Stewart, G. W. (1977). Perturbation bounds for the \(QR\) factorization of a matrix., SIAM Journal on Numerical Analysis 14 509-518. · Zbl 0358.65038
[15] Stone, M. (1974). Cross-validation and multinomial prediction., Biometrika 61 509-515. · Zbl 0292.62025
[16] Storlie, C. B., Bondell, H. D. and Reich, B. J. (2010). A locally adaptive penalty for estimation of functions with varying roughness., To appear in Journal of Computational and Graphical Statistics .
[17] Wahba, G. (1985). A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem., The Annals of Statistics 13 1378-1402. · Zbl 0596.65004
[18] Wang, X., Du, P. and Shen, J. (2013). Smoothing splines with varying smoothing parameter., Biometrika . · Zbl 1279.62099
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.