Density Deconvolution with Small Berkson Errors

TL;DR

This paper investigates density deconvolution with small Berkson errors, revealing conditions under which simple averaging suffices and deriving optimal bandwidths for kernel estimators as errors diminish.

Contribution

It provides a theoretical analysis of density deconvolution with diminishing Berkson errors, identifying when kernel methods are necessary and deriving optimal bandwidth expressions.

Findings

Kernel estimator becomes unnecessary when Berkson error variance exceeds a certain threshold.

Optimal bandwidth expressions are derived for small Berkson errors.

Density of Berkson errors acts as a regularizer in the deconvolution process.

Abstract

The present paper studies density deconvolution in the presence of small Berkson errors, in particular, when the variances of the errors tend to zero as the sample size grows. It is known that when the Berkson errors are present, in some cases, the unknown density estimator can be obtain by simple averaging without using kernels. However, this may not be the case when Berkson errors are asymptotically small. By treating the former case as a kernel estimator with the zero bandwidth, we obtain the optimal expressions for the bandwidth. We show that the density of Berkson errors acts as a regularizer, so that the kernel estimator is unnecessary when the variance of Berkson errors lies above some threshold that depends on the on the shapes of the densities in the model and the number of observations.