Adaptive directional estimator of the density in R^d for independent and mixing sequences

TL;DR

This paper introduces an adaptive multivariate density estimator for stationary sequences that leverages Fourier inversion of a thresholded empirical characteristic function, achieving optimal convergence rates and directional adaptivity without parameter tuning.

Contribution

It proposes a new, parameter-free density estimator that adapts to the regularity and directional features of the unknown density for various mixing sequences.

Findings

Achieves optimal convergence rates up to a logarithmic factor.

Demonstrates adaptation to the regularity of the density and its linear transformations.

Provides an efficient, easy-to-implement calibration procedure using the Euler characteristic.

Abstract

A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, $\alpha$-mixing and $\tau$-mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. In particular, if A is an invertible matrix, if the observations are drawn from X $\in$ R^d , d $\ge$ 1, it achieves the rate implied by the regularity of AX, which may be more regular than X. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas.