Simultaneous estimation of quantile curves using quantile sheets

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P$$\end{document}-splines in form of tensor products of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document}-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.