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.

翻译：通过阈值化经验特征函数的傅里叶反演，本文针对平稳序列构建了一种新的多元密度估计量。该估计量无需依赖于与密度光滑性相关的参数选择，具备直接自适应性。我们建立了适用于独立序列、α-混合序列及τ-混合序列的Oracle不等式，从而在至多对数损失的意义下推导出最优收敛速率。在一般各向异性Sobolev类中，该估计量不仅能自适应于未知密度的正则性，还可实现方向自适应。特别地，对于可逆矩阵A及观测样本X∈R^d（d≥1），该估计量能达到由AX正则性所决定的收敛速率——而AX可能比X具有更高正则性。该估计量易于实现且数值计算高效。其依赖单一参数的校准，为此我们提出了一项创新性数值选择程序，该程序利用阈值区域的欧拉示性数进行参数选取。