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

Abstract

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.