# $\beta$-Divergence loss for the kernel density estimation with bias reduced

## TL;DR

This paper evaluates various bandwidth selection methods for bias-reduced kernel density estimation, focusing on the novel use of $eta$-Divergence loss, through simulations and real econometric data.

## Contribution

It introduces and compares the $eta$-Divergence loss as a new criterion for bandwidth selection in bias-reduced kernel density estimation.

## Key findings

- $eta$-Divergence loss provides a competitive alternative for bandwidth selection.
- Simulation results compare effectiveness of different methods.
- Application to econometric data demonstrates practical utility.

## Abstract

Allthough nonparametric kernel density estimation with bias reduce is nowadays a standard technique in explorative data-analysis, there is still a big dispute on how to assess the quality of the estimate and which choice of bandwidth is optimal. This article examines the most important bandwidth selection methods for kernel density estimation with bias reduce, in particular, normal reference, least squares cross-validation, biased crossvalidation and $\beta$-Divergence loss. Methods are described and expressions are presented. We will compare these various bandwidth selector on simulated data. As an example of real data, we will use econometric data sets CO2 per capita in example 1 and the second