Adaptive Density Estimation Based on Real and Artificial Data

• Activity Number: 548
• Type: Contributed
• Date/Time: Wednesday, August 7, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Tina Felber*+ and Michael Kohler and Adam Krzyzak
• Companies: TU Darmstadt and TU Darmstadt and Concordia University
• Keywords: Density estimation ; $L_1$-error ; nonparametric regression ; consistency ; rate of convergence

Abstract

Let X, X1, X2,... be independent and identically distributed Rd-valued random variables and let m : Rd ! 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 ditribution 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 L1-error of the latter estimate is better than the standard kernel density estimate. Furthermore, a density estimate defined as convex combination of these two estimates is considered and a datadriven choice of its parameters (bandwidths and weight of the convex combination) is proposed and analyzed.