Showing papers by "Frits H. Ruymgaart published in 1999"

Direct density estimation as an ill‐posed inverse estimation problem

Abstract: A possible definition of ill-posedness in statistical estimation is the lack of qualitative robustness. In this sense direct density estimation shares ill-posedness with the more obviously ill-posed indirect density estimation models, of which it is a special case. A general construction pattern for estimators is proposed, based on suitable preconditioning, that works for both direct and indirect density estimation. Special emphasis is on its application to the direct case, where in general it yields delta-sequence estimators. More specifically both kernel and series type estimators are included depending on the choice of preconditioning operator. In particular sinc and other flattop kernel estimators emerge in a natural way.

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

Abstract: 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 corrupted by the Gibbs phenomenon. But this effect can also be remedied using the Cesaro averaging technique known from the direct case. Although the supremum 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 functions. Convergence in this Hausdorff metric entails convergence in the supremum metric if the functions involved are continuous. We obtain a speed of almost sure convergence in the Hausdorff metric for the proposed estimators. This method provides an alternative to an approach from the wavelet or change-point perspective.