Estimation of a Density Using Real and Artificial Data

Let
X
,
X
₁
,
X
₂
, ... be independent and identically distributed R
d
-valued random variables and let
m
: R
d
→ R be a measurable function such that a density
f
of
Y
=
m
(
X
) exists. Given a sample of the distribution of (
X
,
Y
) and additional independent observations of
X
, we are interested in estimating
f
. We apply a regression estimate to the sample of (
X
,
Y
) and use this estimate to generate additional artificial observations of
Y
. Using these artificial observations together with the real observations of
Y
, we construct a density estimate of
f
by using a convex combination of two kernel density estimates. It is shown that if the bandwidths satisfy the usual conditions and if in addition the supremum norm error of the regression estimate converges almost surely faster toward zero than the bandwidth of the kernel density estimate applied to the artificial data, then the convex combination of the two density estimates is
L
₁
-consistent. The performance of the estimate for finite sample size is illustrated by simulated data, and the usefulness of the procedure is demonstrated by applying it to a density estimation problem in a simulation model.