Nonparametric kernel regression with multiple predictors and multiple shape constraints

Pang Du, Christopher F. Parmeter, Jeffrey S. Racine

Research output: Contribution to journalArticlepeer-review

73 Scopus citations


Nonparametric smoothing under shape constraints has recently received much well-deserved attention. Powerful methods have been proposed for imposing a single shape constraint such as monotonicity and concavity on univariate functions. In this paper, we extend the monotone kernel regression method in Hall and Huang (2001) to the multivariate and multi-constraint setting. We impose equality and/or inequality constraints on a nonparametric kernel regression model and its derivatives. A bootstrap procedure is also proposed for testing the validity of the constraints. Consistency of our constrained kernel estimator is provided through an asymptotic analysis of its relationship with the unconstrained estimator. Theoretical underpinnings for the bootstrap procedure are also provided. Illustrative Monte Carlo results are presented and an application is considered.

Original languageEnglish (US)
Pages (from-to)1347-1371
Number of pages25
JournalStatistica Sinica
Issue number3
StatePublished - Jul 2013
Externally publishedYes


  • Hypothesis testing
  • Multivariate kernel estimation
  • Nonparametric regression
  • Shape restrictions

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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