# A combined strategy for multivariate density estimation

## TL;DR

This paper introduces a novel non-linear aggregation method for multivariate density estimation that improves accuracy by considering neighborhoods of estimated level sets, supported by theoretical and simulation results.

## Contribution

It proposes a new density estimation strategy based on level set neighborhoods, addressing computational challenges of existing methods and demonstrating improved mean squared error.

## Key findings

- Lower mean squared error compared to traditional aggregation methods
- Theoretical proof of a Central Limit Theorem for the estimator
- Validated effectiveness through simulation studies

## Abstract

Non-linear aggregation strategies have recently been proposed in response to the problem of how to combine, in a non-linear way, estimators of the regression function (see for instance \cite{biau:16}), classification rules (see \cite{ch:16}), among others. Although there are several linear strategies to aggregate density estimators, most of them are hard to compute (even in moderate dimensions). Our approach aims to overcome this problem by estimating the density at a point $x$ using not just sample points close to $x$ but in a neighborhood of the (estimated) level set $f(x)$. We show, both theoretically and through a simulation study, that the mean squared error of our proposal is smaller than that of the aggregated densities. A Central Limit Theorem is also proven.