Adaptive non-asymptotic confidence balls in density estimation

TL;DR

This paper introduces a method for constructing adaptive, non-asymptotic confidence balls in density estimation using resampling and model selection, ensuring reliable coverage for all sample sizes.

Contribution

It presents a novel approach combining resampling and model selection to create confidence balls that adapt to the underlying density in finite samples.

Findings

Confidence balls have guaranteed coverage for all sample sizes.

The method adapts to the unknown density by selecting optimal approximation spaces.

Resampling estimates improve the accuracy of the confidence sets.

Abstract

We build confidence balls for the common density $s$ of a real valued sample $X_1,...,X_n$. We use resampling methods to estimate the projection of $s$ onto finite dimensional linear spaces and a model selection procedure to choose an optimal approximation space. The covering property is ensured for all $n\geq2$ and the balls are adaptive over a collection of linear spaces.