# zbMATH — the first resource for mathematics

Simultaneous estimation of quantile curves using quantile sheets. (English) Zbl 1443.62107
Summary: The results of quantile smoothing often show crossing curves, in particular, for small data sets. We define a surface, called a quantile sheet, on the domain of the independent variable and the probability. Any desired quantile curve is obtained by evaluating the sheet for a fixed probability. This sheet is modeled by $$P$$-splines in form of tensor products of $$B$$-splines with difference penalties on the array of coefficients. The amount of smoothing is optimized by cross-validation. An application for reference growth curves for children is presented.

##### MSC:
 62G08 Nonparametric regression and quantile regression
##### Keywords:
$$P$$-splines; quantiles; smoothing; tensor product
##### Software:
cobs; Expectreg; quantreg; VGAM; Worm plot
Full Text:
##### References:
 [1] Bassett, GW; Koenker, RW, A note on recent proposals for computing $$l_1$$ estimates, Comput. Stat. Data Anal., 14, 207-211, (1992) · Zbl 0875.62149 [2] Bollaerts, K.; Eilers, PHC; Mechelen, I., Simple and multiple P-splines regression with shape constraints, Br. J. Math. Stat. Psychol., 59, 451-469, (2006) [3] Bondell, HD; Reich, BJ; Wang, H., Noncrossing quantile regression curve estimation, Biometrika, 97, 825-838, (2010) · Zbl 1204.62061 [4] Chernozhukov, V.; Fernandez-Val, I.; Galichon, A., Quantile and probability curves without crossing, Econometrica, 78, 1093-1125, (2010) · Zbl 1192.62255 [5] Cole, TJ; Freeman, JV; Preece, MA, Body mass index reference curves for the UK, 1990, Arch. Dis. Child., 73, 25-29, (1995) [6] Currie, ID; Durban, M.; Eilers, PHC, Generalized linear array models with applications to multidimensional smoothing, J. Roy. Stat. Soc. Ser. B, 68, 259-280, (2006) · Zbl 1110.62090 [7] Dette, H.; Volgushev, S., Non-crossing non-parametric estimates of quantile curves, J. Roy. Stat. Soc. Ser. B, 70, 609-627, (2008) · Zbl 05563361 [8] Eilers, PH; Currie, ID; Durban, M., Fast and compact smoothing on large multidimensional grids, Comput. Stat. Data Anal., 50, 61-76, (2006) · Zbl 1429.62020 [9] Eilers, PHC; Marx, BD, Flexible smoothing with B-splines and penalties, Stat. Sci., 11, 89-121, (1996) · Zbl 0955.62562 [10] Koenker, R.: Quantile Regression. Cambridge University Press, Cambridge (2005) · Zbl 1111.62037 [11] Koenker, R.: (2011) quantreg: Quantile regression. R package version 4.71. http://CRAN.R-project.org/package=quantreg [12] Koenker, R.; Bassett, GW, Regression quantiles, Econometrica, 46, 33-50, (1978) · Zbl 0373.62038 [13] Koenker, R., Ng, P.: (2005) Inequality constrained quantile regression. Special issue on quantile regression and related models. Sankh$$\bar{y}$$a. Indian J. Stat. 67(2):418-440 · Zbl 1193.62023 [14] Newey, WK; Powell, JL, Asymmetric least squares estimation and testing, Econometrica, 55, 819-847, (1987) · Zbl 0625.62047 [15] Ng, PT., Maechler, M.: (2011) cobs: COBS-Constrained B-splines (Sparse matrix based). R package version 1.2-2. http://CRAN.R-project.org/package=cobs [16] Schlossmacher, EJ, An iterative technique for absolute deviations curve fitting, J. Am. Stat. Assoc., 68, 857-859, (1973) · Zbl 0287.62038 [17] Schnabel, SK; Eilers, PHC, Optimal expectile smoothing, Comput. Stat. Data Anal., 53, 4168-4177, (2009) · Zbl 1453.62192 [18] Schnabel, SK., Eilers, P.H.C.: (2011) Expectile sheets for joint estimation of expectile curves (submitted) [19] Shim, J.; Hwang, C.; Seok, KH, Non-crossing quantile regression via doubly penalized kernel machine, Comput. Stat., 23, 83-94, (2009) · Zbl 1223.62050 [20] Sobotka, F., Kneib, T., Schnabel, S., Eilers, P.: (2012) expectreg: Expectile Regression. R package version 0.30. http://CRAN.R-project.org/package=expectreg [21] Soliman, SA; Christensen, GS; Rouhi, A., A new technique for curve fitting based on minimum absolute deviations, Comput. Stat. Data Anal., 6, 341-351, (1988) · Zbl 0726.62112 [22] Soliman, SA; Christensen, GS; Rouhi, AH, A new algorithm for nonlinear $$l_1$$-norm minimization with nonlinear equality constraints, Comput. Stat. Data Anal., 11, 97-109, (1991) · Zbl 0850.62585 [23] Takeuchi, I.; Le, QV; Sears, TD; Smola, AJ, Nonparametric quantile estimation, J. Mach. Learn. Res., 7, 1231-1264, (2006) · Zbl 1222.68316 [24] Buuren, S.; Fredriks, AM, Worm plot: a simple diagnostic device for modeling growth reference curves, Stat. Med., 20, 1259-1277, (2001) [25] WHO Multicentre Growth Reference Study Group: WHO child growth standards: growth velocity based on weight, length and head circumference: methods and developments, Tech. rep. World Health Organization, Geneva (2009) [26] Yee, TW (2011) VGAM: vector generalized linear and additive models. R package version 0.8-3. http://CRAN.R-project.org/package=VGAM
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.