Estimation of Intensities of Burst - Like Poisson Processes Using HaarWavelets

I present a method for producing estimates of the intensity function of certain `burst-like' inhomogeneous Poisson processes, based on the shrinkage of Haar wavelet coe cients of the observed counts. The Haar basis is a natural wavelet basis in which to work in this context, and I derive thresholds for shrinkage estimation based on the distribution of the coe cients. The translation-invariant de-noising approach of Donoho and Coifman (1995) is used in conjunction with these level-dependent thresholds to yield smooth estimates, without the usual `staircase' structure associated with Haar wavelets. This work is motivated by recent e orts in astronomy to model the intensity functions underlying gamma-ray bursts. It is demonstrated that the method proposed herein (TIPSH) yields sharper estimates of the detail structure in these signals than those obtained through an analogous version of the standard adaptation of wavelet shrinkage for Poisson counts, based on the square-root transformation.