Adaptive density estimation based on real and artificial data

Let X, X-1, X-2, horizontal ellipsis be independent and identically distributed Double-struck capital R-d-valued random variables and let m:Double-struck capital R-d -> Double-struck capital R be a measurable function such that a density f of Y=m(X) exists. The problem of estimating f based on a sample of the distribution of (X,Y) and on additional independent observations of X is considered. Two kernel density estimates are compared: the standard kernel density estimate based on the y-values of the sample of (X,Y), and a kernel density estimate based on artificially generated y-values corresponding to the additional observations of X. It is shown that under suitable smoothness assumptions on f and m the rate of convergence of the L-1 error of the latter estimate is better than that of the standard kernel density estimate. Furthermore, a density estimate defined as convex combination of these two estimates is considered and a data-driven choice of its parameters (bandwidths and weight of the convex combination) is proposed and analysed.