Lower bounds on estimator performance for energy-invariant parameters of multidimensional Poisson processes

Using rate distortion theory, lower bounds are developed for the mean-square error of estimates of a random parameter of an M-dimensional inhomogeneous Poisson process with respect to which the energy, i.e. the average number of points, is invariant. The bounds are derived without stringent assumptions on either the form of the intensity or the prior distribution of the parameter, and they can handle random nuisance parameters. The derivation makes use of a side-information averaging principle applied to the distortion-rate function and a maximum-entropy property of energy-constrained Poisson processes. Under the additional assumption of conditional entropy invariance of the point process with respect to the parameter of interest, an explicit bound is given which depends on the information discrimination between the inhomogeneous conditionally Poisson process and a nearly homogeneous Poisson process. The application of the explicit bound is illustrated through a treatment of the problems of time-shift estimation and relative time-shift estimation for Poisson streams