Nonparametric confidence bands in deconvolution density estimation

Uniform confidence bands for densities f via nonparametric kernel estimates were first constructed by Bickel and Rosenblatt [Ann. Statist. 1, 1071–1095]. In this paper this is extended to confidence bands in the deconvolution problem g = f ∗ ψ for an ordinary smooth error density ψ. Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (L∞-distance) between a deconvolution kernel estimator f̂ and f . Further consistency of the simple nonparametric bootstrap is proved. For our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth. For practical purposes we propose a new data-driven bandwidth selector based on heuristic arguments, which aims Adress for correspondence: Dr. Nicolai Bissantz, Faculty of Mathematics, Ruhr-University Bochum, Universitätsstraße 150, Mathematik III NA 3/70, D-44780 Bochum, Germany, email: [email protected], Fon: +49/234/32–14559, Fax: +49/234/32–23291