Seminar Statistics for Life Sciences: Johannes Schmidt-Hieber

04/16/2012

11:15-12:15

Room M-616, De Boelelaan 1081 6th floor M corridor

Confidence Statements for Qualitative Features in Deconvolution

Johannes Schmidt-Hieber

Fetsje Bijma

Faculty of Science

Sciences

Conference / Symposium / Seminar

abstract

Suppose that we observe data from a deconvolution model, that is, we observe an i.i.d. sample from an unknown distribution under additive noise. In many practical problems the main interest lies not in pointwise reconstruction of the true density but rather in answering qualitative questions, for instance about the number of modes or the regions of increase and decrease.

In this talk, we derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. Important examples covered within this framework are to test for local monotonicity or local convexity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. Theoretically we derive convergence rates for the distance between the multiscale statistic and a distribution-free approximation and study the power of the constructed tests. In a second part, we illustrate our method by numerical simulations. A major consequence of our work is that the detection of qualitative features of a density inĀ  a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow. This is joint work with Axel Munk (G\"ottingen) and Lutz D\"umbgen (Bern).