Density estimation using real and artificial data

Let X, X₁, X₂, ... be independent and identically distributed ℝ^d-valued random variables and let m : ℝ^d → ℝ be an unknown measurable function such that a density f of Y = m(X) exists. In this paper we consider estimating f based on i.i.d. sample (X₁, Y₁);...; (X_n, Y_n) of (X, Y) and on additional independent observations of X. We compare 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 L₁ 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 the bandwidths and the weight of the convex combination is proposed and investigated.