×

Additive models for quantile regression: model selection and confidence bands. (English) Zbl 1236.62031

Summary: Additive models for conditional quantile functions provide an attractive framework for nonparametric regression applications focusing on the features of the response beyond its central tendency. Total variation roughness penalities can be used to control the smoothness of the additive components much as squared Sobelev penalties are used for classical \(L_{2}\) smoothing splines. We describe a general approach to estimation and inference for additive models of this type. We focus attention primarily on the selection of the smoothing parameters and on the construction of confidence bands for the nonparametric components. Both pointwise and uniform confidence bands are introduced; the uniform bands are based on the H. Hotelling [Am. J. Math. 61, 440–460 (1939: Zbl 0020.38302)] tube approach. Some simulation evidence is presented to evaluate finite sample performance and the methods are also illustrated with an application to modeling childhood malnutrition in India.

MSC:

62G08 Nonparametric regression and quantile regression
62G15 Nonparametric tolerance and confidence regions
65C60 Computational problems in statistics (MSC2010)

Citations:

Zbl 0020.38302
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Belloni, A. and Chernozhukov, V. (2009)., L 1 -penalized quantile regression in high-dimensional sparse models. Available at . · Zbl 1175.65015
[2] Breiman, L. and Friedman, J. (1985). Estimating optimal transformations for multiple regression and correlation., Journal of the American Statistical Association 80 , 580-598. · Zbl 0594.62044
[3] Davies, P. L. and Kovac, A. (2001). Local extremes, runs, strings and multiresolution., Ann. Statist. 29 , 1-48. · Zbl 1029.62038
[4] Delbaere, I., Vansteelandt, S., Bacquer, D. D., Verstraelen, H., Gerris, J., Sutter, P. D. and Temmerman, M. (2007). Should we adjust for gestational age when analysing birth weights? The use of z-scores revisited., Human Reproduction 22 , 2080-2083.
[5] Fenske, N., Kneib, T. and Hothorn, T. (2011). Identifying risk factors for severe childhood malnutrition by boosting additive quantile regression., Journal of the American Statistical Association 106 , 494-510. · Zbl 1232.62146
[6] Hastie, T. and Tibshirani, R. (1986). Generalized additive models., Statistical Science 1 , 297-310. · Zbl 0645.62068
[7] Hastie, T. and Tibshirani, R. (1990)., Generalized Additive Models . London: Chapman and Hall. · Zbl 0747.62061
[8] Hotelling, H. (1939). Tubes and spheres in, n -space and a class of statistical problems. American Journal of Mathematics 61 , 440-460. · Zbl 0020.38302
[9] Johansen, S. and Johnstone, I. M. (1990). Hotelling’s theorem on the volume of tubes: Some illustrations in simultaneous inference and data analysis., The Annals of Statistics 18 , 652-684. · Zbl 0723.62018
[10] Knowles, M. (1987). Simultaneous confidence bands for random functions. Ph.D. thesis, Stanford, Univ.
[11] Koenker, R. (2005)., Quantile Regression . London: Cambridge Univ. Press. · Zbl 1111.62037
[12] Koenker, R. (2010). quantreg: Quantile Regression, v4.45. Available at, .
[13] Koenker, R. and Mizera, I. (2004). Penalized triograms: Total variation regularization for bivariate smoothing., J. R. Stat. Soc. Ser. B Stat. Methodol. 66 , 145-163. · Zbl 1064.62038
[14] Koenker, R. and Ng, P. (2005). A Frisch-Newton algorithm for sparse quantile regression., Mathematicae Applicatae Sinica 21 , 225-236. · Zbl 1097.62028
[15] Koenker, R., Ng, P. and Portnoy, S. (1994). Quantile smoothing splines., Biometrika 81 , 673-680. · Zbl 0810.62040
[16] Krivobokova, T., Kneib, T. and Claeskens, G. (2011). Simultaneous confidence bands for penalized spline estimators., Journal of the American Statistical Association 105 , 852-863. · Zbl 1392.62094
[17] Loader, C. (2010). locfit: Local regression, likelihood and density estimation, v1.5-5. Available at, .
[18] Machado, J. (1993). Robust model selection and M-estimation., Econometric Theory 9 , 478-493. · Zbl 04515826
[19] Meyer, M. and Woodroofe, M. (2000). On the degrees of freedom in shape-restricted regression., The Annals of Statistics 28 , 1083-1104. · Zbl 1105.62340
[20] Naiman, D. (1986). Conservative confidence bands in curvilinear reression., The Annals of Statistics 14 , 896-906. · Zbl 0607.62077
[21] Nychka, D. (1983). Bayesian Confidence Intervals for smoothing splines., Journal of the American Statistical Association 83 , 1134-1143.
[22] Pötscher, B. and Leeb, H. (2009). On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD and thresholding., Journal of Multivariate Analysis 100 , 2065-2082. · Zbl 1170.62046
[23] Powell, J. L. (1991). Estimation of monotonic regression models under quantile restrictions. In, Nonparametric and Semiparametric Methods in Econometrics (W. Barnett, J. Powell and G. Tauchen, eds.) 357-384. Cambridge: Cambridge Univ. Press. · Zbl 0754.62023
[24] Rudin, L., Osher, S. and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms., Physica D 60 , 259-268. · Zbl 0780.49028
[25] Ruppert, D., Wand, M. and Carroll, R. J. (2003)., Semiparametric Regression . Cambridge: Cambridge Univ. Press. · Zbl 1038.62042
[26] Schwarz, G. (1978). Estimating the dimension of a model., The Annals of Statistics 6 , 461-464. · Zbl 0379.62005
[27] Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields., The Annals of Probability 21 , 34-71. · Zbl 0772.60038
[28] Sun, J. and Loader, C. (1994). Simultaneous confidence bands for linear regression and smoothing., The Annals of Statistics 22 , 1328-1347. · Zbl 0817.62057
[29] Sun, J., Loader, C. and McCormick, W. (2000). Confidence bands in generalized linear models., The Annals of Statistics 28 , 429-460. · Zbl 1106.62343
[30] Tibshirani, R., Saunders, M., Rosset, S., Zhu, J. and Knight K. (2005). Sparsity and smoothness via the fused lasso., Journal of the Royal Statistical Society B 67 , 91-108. · Zbl 1060.62049
[31] Wahba, G. (1983). Bayesian “Confidence Intervals” for the cross-validated smoothing spline., Journal of the Royal Statistical Society B 45 , 133-150. · Zbl 0538.65006
[32] Wand, M. P. and Ormerod, J. T. (2008). On semiparametric regression with O’Sullivan penalized splines., Aust. N. Z. J. Stat. 50 , 179-198. · Zbl 1146.62030
[33] Weyl, H. (1939). On the Volume of Tubes., Amer. J. Math. 61 , 461-472. · Zbl 0021.35503
[34] Wood, S. (2006)., Generalized Additive Models: An Introduction with R . Boca Raton, FL: Chapman-Hall. · Zbl 1087.62082
[35] Wood, S. (2010). mgcv: GAMs with GCV/AIC/REML smoothness estimation and GAMMs by PQL. Available at, .
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.