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 Cesàro 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.

Keywords:

• Irregular input signals
• Gibbs phenomenon
• Hausdorff metric
• Cesàro averaging
• speed of convergence

^*Work supported by NSF grant DMS 95-04485.

^†Work supported by Texas ARP grant 003644-116 and Work supported by NSF grant DMS 95-04485.

^‡Corresponding author.

^*Work supported by NSF grant DMS 95-04485.

^†Work supported by Texas ARP grant 003644-116 and Work supported by NSF grant DMS 95-04485.

^‡Corresponding author.

Notes

^*Work supported by NSF grant DMS 95-04485.

^†Work supported by Texas ARP grant 003644-116 and Work supported by NSF grant DMS 95-04485.

^‡Corresponding author.