Original Paper

AStA Advances in Statistical Analysis

, Volume 97, Issue 1, pp 77-87

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Simultaneous estimation of quantile curves using quantile sheets

  • Sabine K. SchnabelAffiliated withBiometris, Wageningen University and Research Centre Email author 
  • , Paul H. C. EilersAffiliated withDepartment of Biostatistics, Erasmus Medical Center


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.


\(P\)-splines Quantiles Smoothing Tensor product