I-SMOOTH: Iteratively smoothing piecewise-constant Poisson-process rate functions

Piecewise-constant Poisson process rate functions are easy to estimate and provide easy random-process generation. When the true rate function is continuous, however, a piecewise-constant approximation is sometimes unacceptably crude. Given a non-negative piecewise-constant rate function, we discuss SMOOTH (Smoothing via Mean-constrained Optimized-Objective Time Halving), a quadratic optimization formulation that yields a smoother non-negative piecewise-constant rate function having twice as many time intervals, each of half the length. I-SMOOTH (Iterated SMOOTH) iterates the SMOOTH formulation to create a sequence of piecewise-constant rate functions having an asymptotic continuous rate function. We consider two contexts: finite-horizon and cyclic. We develop a sequence of computational simplifications for SMOOTH, moving from numericallyminimizing the quadratic objective function, to numerically computing a matrix inverse, to a closed-form matrix inverse obtained as finite sums, to decision variables that are linear combinations of the given rates, and to simple approximations.