×

Simultaneous estimation of quantile regression functions using B-splines and total variation penalty. (English) Zbl 1507.62085

Summary: We consider the problem of simultaneously estimating a finite number of quantile functions with B-splines and the total variation penalty. For the implementation of simultaneous quantile function estimators, we develop a new coordinate descent algorithm taking into account a special structure of the total variation penalty determined by B-spline coefficients. The entire paths of solution paths for several quantile function estimators and tuning parameters can be efficiently computed using the coordinate descent algorithm. We also consider non-crossing quantile function estimators having additional constraints at the knots of spline functions. Numerical studies using both simulated and real data sets are provided to illustrate the performance of the proposed method. For a theoretical result, we prove that the proposed the quantile regression function estimators achieve the minimax rate under regularity conditions.

MSC:

62-08 Computational methods for problems pertaining to statistics
62G08 Nonparametric regression and quantile regression
62G07 Density estimation

Software:

COBS; QICD
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bachrach, L. K.; Hastie, T.; Wang, M.; Narasimhan, B.; Marcus, R., Bone mineral acquisition in healthy asian, hispanic, black, and caucasian youth: a longitudinal study 1, J. Clin. Endocrinol. Metab., 84, 12, 4702-4712 (1999)
[2] Bollaerts, K.; Eilers, P. H.C.; Aerts, M., Quantile regression with monotonicity restrictions using p-splines and the \(L_1\)-norm, Stat. Model., 6, 3, 189-207 (2006) · Zbl 07257134
[3] Bondell, H. D.; Reich, B. J.; Wang, H., Noncrossing quantile regression curve estimation, Biometrika, 825-838 (2010) · Zbl 1204.62061
[4] Breheny, P.; Huang, J., Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection, Ann. Appl. Stat., 5, 1, 232 (2011) · Zbl 1220.62095
[5] Breiman, L.; Peters, S., Comparing automatic smoothers (a public service enterprise), Int. Stat. Rev., 271-290 (1992) · Zbl 0775.62089
[6] Chaudhuri, P., Global nonparametric estimation of conditional quantile functions and their derivatives, J. Multivariate Anal., 39, 2, 246-269 (1991) · Zbl 0739.62028
[7] Dembo, A.; Zeitouni, O., Large deviations techniques and applications, Large Deviations Techniques and Applications 38 (1998) · Zbl 0896.60013
[8] Doksum, K.; Koo, J. Y., On spline estimators and prediction intervals in nonparametric regression, Comput. Statist. Data Anal., 35, 1, 67-82 (2000) · Zbl 1142.62340
[9] Donoho, D. L.; Johnstone, I. M., Ideal spatial adaption by wavelet shirinkage, Biometrika, 80, 425-455 (1994) · Zbl 0815.62019
[10] Friedman, J.; Hastie, T.; Hofling, H.; Tibshirani, R., Pathwise coordinate optimazation, Ann. Statist., 1, 2, 302-332 (2007) · Zbl 1378.90064
[11] Friedman, J.; Hastie, T.; Tibshirani, R., (The Elements of Statistical Learning. The Elements of Statistical Learning, Springer Series in Statistics, vol. 1 (2001), Springer) · Zbl 0973.62007
[12] He, X.; Ng, P., COBS: qualitatively constrained smoothing via linear programming, Comput. Stat., 14, 3, 315-338 (1999) · Zbl 0941.62037
[13] Jhong, J.; Koo, J.; Lee, S., Penalized b-spline estimator for regression functions using total variation penalty, J. Statist. Plann. Inference, 184, 77-93 (2017) · Zbl 1395.62080
[14] Koenker, R.; Bassett Jr, G., Regression quantiles, Econometrica, 33-50 (1978) · Zbl 0373.62038
[15] Koenker, R.; Mizera, I., Penalized triograms: total variation regularization for bivariate smoothing, J. R. Stat. Soc. Ser. B Stat. Methodol., 66, 1, 145-163 (2004) · Zbl 1064.62038
[16] Koenker, R.; Ng, P.; Portnoy, S., Quantile smoothing splines, Biometrika, 81, 4, 673-680 (1994) · Zbl 0810.62040
[17] Loubes, J. M.; Van De Geer, S., Adaptive estimation with soft thresholding penalties, Stat. Neerl., 56, 4, 453-478 (2002) · Zbl 1090.62534
[18] Massart, P., Some applications of concentration inequalities to statistics, (Annales-Faculte des Sciences Toulouse Mathematiques. Vol. 9 (2000), Université Paul Sabatier), 245-303 · Zbl 0986.62002
[19] Nason, G. P.; Silverman, B. W., The discrete wavelet transform in s, J. Comput. Graph. Statist., 3, 2, 163-191 (1994)
[20] Ng, P.; Maechler, M., A fast and efficient implementation of qualitatively constrained quantile smoothing splines, Stat. Model., 7, 4, 315-328 (2007) · Zbl 1486.62118
[21] Peng, B.; Wang, L., An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression, J. Comput. Graph. Statist., 24, 3, 676-694 (2015)
[22] Schwarz, G., Estimating the dimension of a model, Ann. Statist., 6, 2, 461-464 (1978) · Zbl 0379.62005
[23] Takeuchi, I.; Furuhashi, T., Non-crossing quantile regressions by svm, (Neural Networks, 2004. Proceedings. 2004 IEEE International Joint Conference on. Vol. 1 (2004), IEEE), 401-406
[24] Takeuchi, I.; Le, Q. V.; Sears, T. D.; Smola, A. J., Nonparametric quantile estimation, J. Mach. Learn. Res., 7, Jul, 1231-1264 (2006) · Zbl 1222.68316
[25] Tibshirani, R., Regression shrinkage and selection via the lasso, J. R. Stat. Soc., 58, 1, 267-288 (1996) · Zbl 0850.62538
[26] Tibshirani, R.; Saunders, M., Sparsity and smoothness via the fused lasso, J. R. Stat. Soc., 67, 1, 91-108 (2005) · Zbl 1060.62049
[27] van der Vaart, A.; Wellner, J., (Weak Convergence and Empirical Processes: With Applications to Statistics. Weak Convergence and Empirical Processes: With Applications to Statistics, Springer Series in Statistics (1996), Springer) · Zbl 0862.60002
[28] Welsh, A., Robust estimation of smooth regression and spread functions and their derivatives, Statist. Sinica, 347-366 (1996) · Zbl 0884.62047
[29] Wu, T. T.; Lange, K., Coordinate descent algorithms for lasso penalized regression, Ann. Appl. Stat., 224-244 (2008) · Zbl 1137.62045
[30] Yi, C.; Huang, J., Semismooth newton coordinate descent algorithm for elastic-net penalized huber loss regression and quantile regression, J. Comput. Graph. Statist., 1-11 (2017)
[31] Zhou, S.; Shen, X.; Wolfe, D. A., Local asymptotics for regression splines and confidence regions, Ann. Statist., 26, 5, 1760-1782 (1998) · Zbl 0929.62052
[32] Zou, H.; Hastie, T., Regularization and variable selection via the elastic net, J. R. Stat. Soc., 67, 2, 301-320 (2005) · Zbl 1069.62054
[33] Zou, H.; Yuan, M., Composite quantile regression and the oracle model selection theory, Ann. Statist., 1108-1126 (2008) · Zbl 1360.62394
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.