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Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds. (English) Zbl 0927.62081
Summary: Wavelet shrinkage estimation has been found to be a powerful tool for the nonparametric estimation of spatially variable phenomena. Most work in this area to date has concentrated primarily on the use of wavelet shrinkage techniques in contexts where the data are modeled as observations of a signal plus additive Gaussian noise. In this paper, I introduce an approach to estimating intensity functions for a certain class of “burst-like” Poisson processes using wavelet shrinkage. The proposed method is based on the shrinkage of wavelet coefficients of the original count data, as opposed to the current approach of pre-processing the data using Anscombe’s square root transform and working with the resulting data in a Gaussian framework.
“Corrected” versions of the usual Gaussian-based shrinkage thresholds are used. The 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 performance of the new method is examined, and compared to that of the pre-processing approach, in the context of an application to an astronomical gamma-ray burst signal.

62M07 Non-Markovian processes: hypothesis testing
62G07 Density estimation
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems