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We derive multiscale statistics for deconvolution in order to detectqualitative features of the unknown density. An important example coveredwithin this framework is to test for local monotonicity on all scalessimultaneously. We investigate the moderately ill-posed setting, where theFourier transform of the error density in the deconvolution model is ofpolynomial decay. For multiscale testing, we consider a calibration, motivatedby the modulus of continuity of Brownian motion. We investigate the performanceof our results from both the theoretical and simulation based point of view. Amajor consequence of our work is that the detection of qualitative features ofa density in a deconvolution problem is a doable task although the minimaxrates for pointwise estimation are very slow.

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