Fully Data-driven Normalized and Exponentiated Kernel Density Estimator with Hyv\"arinen Score

TL;DR

This paper proposes a novel data-driven kernel density estimator that uses an exponentiated form and Hyv"arinen score for hyperparameter tuning, effectively handling multi-modal densities and outliers.

Contribution

It introduces a new exponentiated kernel density estimator with data-driven hyperparameter tuning via Hyv"arinen score, providing flexible and robust density estimation.

Findings

Effective for multi-modal densities

Handles outliers well

Outperforms traditional methods in simulations

Abstract

We introduce a new deal of kernel density estimation using an exponentiated form of kernel density estimators. The density estimator has two hyperparameters flexibly controlling the smoothness of the resulting density. We tune them in a data-driven manner by minimizing an objective function based on the Hyv\"arinen score to avoid the optimization involving the intractable normalizing constant due to the exponentiation. We show the asymptotic properties of the proposed estimator and emphasize the importance of including the two hyperparameters for flexible density estimation. Our simulation studies and application to income data show that the proposed density estimator is appealing when the underlying density is multi-modal or observations contain outliers.