Bivariate quantile smoothing splines

Xuming He, Pin Ng, Stephen Portnoy

Research output: Contribution to journalArticlepeer-review

84 Scopus citations


It has long been recognized that the mean provides an inadequate summary whereas the set of quantiles can supply a more complete description of a sample. We introduce bivariate quantile smoothing splines, which belong to the space of bilinear tensor product splines, as non-parametric estimators for the conditional quantile functions in a two-dimensional design space. The estimators can be computed by using standard linear programming techniques and can further be used as building-blocks for conditional quantile estimations in higher dimensions. For moderately large data sets, we recommend penalized bivariate B-splines as approximate solutions. We use real and simulated data to illustrate the methodology proposed.

Original languageEnglish (US)
Pages (from-to)537-550
Number of pages14
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Issue number3
StatePublished - 1998


  • Conditional quantile
  • Linear program
  • Nonparametric regression
  • Robust regression
  • Schwarz information criterion
  • Tensor product spline

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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