Speed of convergence in the hausdorff metric for estimators of irregular mixing densities

In this paper we consider a noisy deconvolution problem where the signal to be recovered is irregular. Like in the ordinary, direct, estimation models also in the present indirect set-up the approximation or estimate is corru pted by the Gibbs phenomenon. But this effect can also be remedied using th e Cesaro averaging technique known from the direct case. Although the supre mum norm itself is unsuitable it seems adequate to asses the quality of the estimator in a metric related to it. Here we propose the metric defined by the Hausdorff distance between the extended, closed graphs of two function s. Convergence in this Hausdorff metric entails convergence in the supremum metric if the functions involved are continuous. We obtain a speed of almo st sure convergence in the Hausdorff metric for the proposed estimators. Th is method provides an alternative to an approach from the wavelet or change -point perspective.