Some thoughts on the asymptotics of the deconvolution kernel density estimator

TL;DR

This paper compares the finite sample performance of the deconvolution kernel density estimator with its asymptotic predictions, revealing discrepancies at low noise levels and better alignment at higher noise levels, motivating further study of error variance tending to zero.

Contribution

It provides a simulation-based comparison of finite sample and asymptotic behaviors of the deconvolution kernel density estimator under supersmooth noise conditions.

Findings

Asymptotic theory poorly matches finite sample performance at low noise levels.

Better agreement between theory and simulation occurs at higher noise levels.

Results motivate studying deconvolution with error variance approaching zero as sample size increases.

Abstract

Via a simulation study we compare the finite sample performance of the deconvolution kernel density estimator in the supersmooth deconvolution problem to its asymptotic behaviour predicted by two asymptotic normality theorems. Our results indicate that for lower noise levels and moderate sample sizes the match between the asymptotic theory and the finite sample performance of the estimator is not satisfactory. On the other hand we show that the two approaches produce reasonably close results for higher noise levels. These observations in turn provide additional motivation for the study of deconvolution problems under the assumption that the error term variance $\sigma^2\to 0$ as the sample size $n\to\infty.$