Pointwise adaptive estimation of a multivariate density under independence hypothesis

TL;DR

This paper introduces a data-driven, adaptive kernel estimator for multivariate density estimation that automatically adjusts to independence structures, reducing the curse of dimensionality and achieving minimax optimality.

Contribution

It proposes a novel pointwise adaptive estimation method that automatically detects independence structures, improving accuracy in high-dimensional density estimation.

Findings

Estimator is minimax optimal.

Automatic adaptation to independence reduces dimensionality effects.

Provides a data-driven selection rule with oracle inequality.

Abstract

In this paper, we study the problem of pointwise estimation of a multivariate density. We provide a data-driven selection rule from the family of kernel estimators and derive for it a pointwise oracle inequality. Using the latter bound, we show that the proposed estimator is minimax and minimax adaptive over the scale of anisotropic Nikolskii classes. It is important to emphasize that our estimation method adjusts automatically to eventual independence structure of the underlying density. This, in its turn, allows to reduce significantly the influence of the dimension on the accuracy of estimation (curse of dimensionality). The main technical tools used in our considerations are pointwise uniform bounds of empirical processes developed recently in Lepski [Math. Methods Statist. 22 (2013) 83-99].