Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds

Wavelet shrinkage estimation has been found to be a powerful tool for the n on-parametric estimation of spatially variable phenomena. Most work in this area to date has concentrated primarily on the use of wavelet shrinkage te chniques in contexts where the data are modeled as observations of a signal plus additive, Gaussian noise. In this paper, I introduce an approach to e stimating intensity functions for a certain class of "burst-like" Poisson p rocesses using wavelet shrinkage. The proposed method is based on the shrin kage of wavelet coefficients of the original count data, as opposed to the current approach of pre-processing the data using Anscombe's square root tr ansform and working with the resulting data in a Gaussian framework. "Corre cted" versions of the usual Gaussian-based shrinkage thresholds are used. T he corrections explicitly account for effects of the first few cumulants of the Poisson distribution on the tails of the coefficient distributions. A large deviations argument is used to justify these corrections. The perform ance of the new method is examined, and compared to that of the pre-process ing approach, in the context of an application to an astronomical gamma-ray burst signal.