  It is common, in deconvolution problems, to assume that the measurement
errors are identically distributed. In many real-life applications, however,
this condition is not satisfied and the deconvolution estimators developed for
homoscedastic errors become inconsistent. In this paper, we introduce a kernel
estimator of a density in the case of heteroscedastic contamination. We
establish consistency of the estimator and show that it achieves optimal rates
of convergence under quite general conditions. We study the limits of
application of the procedure in some extreme situations, where we show that, in
some cases, our estimator is consistent, even when the scaling parameter of the
error is unbounded. We suggest a modified estimator for the problem where the
distribution of the errors is unknown, but replicated observations are
available. Finally, an adaptive procedure for selecting the smoothing parameter
is proposed and its finite-sample properties are investigated on simulated
examples.
