HIGH-RESOLUTION INVERSION OF THE DISCRETE POISSON AND BINOMIAL TRANSFORMATIONS

The discrete Poisson transformation involved in finite Poisson mixtures arises in many applications, for example in relating observed fluctuations in photon counts to unobserved fluctuations in the number of molecules present, in the analysis of egg counts in zoology, of plankton abundances in fisheries research, of death notice frequencies or of comet frequencies to name a few. The binomial transformation, in addition to its many applications, serves to approximate the Poisson transformation. Procedures such as the maximum likelihood method (MLM) and the moment method (mm) are frequently used to invert these transforms. For the limited data case we consider here, the low signal-to-noise ratio in the data necessitates some modification of existing methods. We propose a filtering procedure based on regularized Fourier transform estimation, followed by high-resolution eigenvector-based spectral estimation. Examples are given which compare this approach with MM and MLM. We find improved performance in the estimation of parameters for the discrete model, particularly for the cases of low data set size and reduced signal-to-noise ratios.