Kernel Density Estimation with Berkson Error

TL;DR

This paper develops kernel density estimators for the convolution of a true density with known Berkson error, compares bandwidth selection methods, and introduces a data-driven bandwidth estimator with applications in epidemiology.

Contribution

It introduces a new approach for bandwidth selection in Berkson error density estimation and analyzes its performance both asymptotically and through simulations.

Findings

Optimal bandwidth depends on the error structure.

Smoothing is crucial when the error density is concentrated near zero.

The proposed data-driven estimator performs well on NO₂ exposure data.

Abstract

Given a sample $\{X_i\}_{i=1}^n$ from $f_X$, we construct kernel density estimators for $f_Y$, the convolution of $f_X$ with a known error density $f_{\epsilon}$. This problem is known as density estimation with Berkson error and has applications in epidemiology and astronomy. Little is understood about bandwidth selection for Berkson density estimation. We compare three approaches to selecting the bandwidth both asymptotically, using large sample approximations to the MISE, and at finite samples, using simulations. Our results highlight the relationship between the structure of the error $f_{\epsilon}$ and the optimal bandwidth. In particular, the results demonstrate the importance of smoothing when the error term $f_{\epsilon}$ is concentrated near 0. We propose a data--driven bandwidth estimator and test its performance on NO$_2$ exposure data.