Multiscale Methods for Shape Constraints in Deconvolution: Confidence Statements for Qualitative Features

• Activity Number: 4
• Type: Invited
• Date/Time: Sunday, August 4, 2013 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Axel E. Munk*+ and Johannes Anselm Schmidt-Hieber and Lutz Dümbgen
• Companies: University of Goettingen and ENSAE, Paris and Institute of Mathematical Statistics and Actuarial Science, University of Bern
• Keywords: deconvolution ; multiscale methods ; shape constraints ; pseudo differential operator ; monotone estimation ; qualitative inference

Abstract

We derive multiscale statistics for deconvolution in order to detect qualitative features of an unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. other examples are bumps or segments of convexity/concavity. For multiscale testing, we consider a specific calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view and some graphical tools are provided to assess simultaneously qualitative structural features of the density. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation or for confidence bands are extremely slow.