  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.
