17.1 Deconvolution density and regression estimates
Deconvolution kernel estimates have been described and extensively
discussed in the context of estimating a probability density from independent
and identically distributed data (Stefansky and Carroll; 1990; Carroll and Hall; 1988).
To explain the basic idea behind this type of
estimates we consider the deconvolution problem first. Let
be
independent and identically distributed real random variables with density
which we want to estimate. We do not, however, observe the
directly but only with additive errors
.
Let us assume that the
as well are
independent and identically distributed with density
and independent
of the
Hence, the available data are
To be able to identify the distribution of the
from the errors
at all, we have to assume that
is known. The
density of the observations
is just the convolution of
with
:
We can therefore try to estimate
by a common kernel estimate and
extract an estimate for
out of it. This kind of deconvolution
operation is preferably performed in the frequency domain, i.e. after applying a Fourier
transform. As the subsequent inverse Fourier
transform includes already a smoothing part we can start with the empirical
distribution of
instead of a smoothed version of it. In
detail, we calculate the Fourier transform or characteristic function of the
empirical law of
, i.e. the sample characteristic function
Let
denote the (known) characteristic function of the
Furthermore, let
be a common kernel function, i.e. a nonnegative continuous function which is
symmetric around 0 and integrates up to 1:
and let
be its Fourier transform. Then, the deconvolution kernel density estimate of
is defined as
The name of this estimate is explained by the fact that it may be written
equivalently as a kernel density estimate
with deconvolution kernel
depending explicitly on the smoothing parameter
. Based on this kernel
estimate for probability densities, Fan and Truong (1993) considered the
analogous deconvolution kernel regression estimate defined as
This Nadaraya-Watson-type estimate is consistent for the regression function
in an errors-in-variables regression model
where
are independent identically distributed zero-mean
random variables independent of the
which are chosen as
above. The
are observed, and the probability density of the
has to be known.