• Bernoulli
  • Volume 25, Number 1 (2019), 549-583.

Limit properties of the monotone rearrangement for density and regression function estimation

Dragi Anevski and Anne-Laure Fougères

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The monotone rearrrangement algorithm was introduced by Hardy, Littlewood and Pólya as a sorting device for functions. Assuming that $x$ is a monotone function and that an estimate $x_{n}$ of $x$ is given, consider the monotone rearrangement $\hat{x}_{n}$ of $x_{n}$. This new estimator is shown to be uniformly consistent as soon as $x_{n}$ is. Under suitable assumptions, pointwise limit distribution results for $\hat{x}_{n}$ are obtained. The framework is general and allows for weakly dependent and long range dependent stationary data. Applications in monotone density and regression function estimation are detailed. Asymptotics for rearrangement estimators with vanishing derivatives are also obtained in these two contexts.

Article information

Bernoulli, Volume 25, Number 1 (2019), 549-583.

Received: August 2016
Revised: October 2017
First available in Project Euclid: 12 December 2018

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density estimation dependence limit distributions monotone rearrangement regression function estimation


Anevski, Dragi; Fougères, Anne-Laure. Limit properties of the monotone rearrangement for density and regression function estimation. Bernoulli 25 (2019), no. 1, 549--583. doi:10.3150/17-BEJ998.

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Supplemental materials

  • Supplement to “Limit properties of the monotone rearrangement for density and regression function estimation”. The supplemental article Anevski and Fougères [1] provides in a first section some technical results on maximal bounds for the rescaled version partial sum and empirical process; it gives further conditions under which Assumption 2 holds, with application to the density and regression function estimation cases, stated in Appendix B, as well as all proofs. Furthermore, Section 2 of Anevski and Fougères [1] contains a simulation study that illustrates the finite sample behaviour of our estimator, and compare it to other estimators that are considered in the paper of Birke and Dette [6].