Experimental evaluation of a procedure for estimating nonhomogeneous Poisson processes having cyclic behavior

This paper summarizes an experimental evaluation of a procedure for estimating a nonhomogeneous Poisson process having an exponential rate function, where the exponent may include both polynomial and trigonometric components. Maximum likelihood estimates of the unknown continuous parameters of the rate function are obtained numerically, and the degree of the polynomial rate component is determined by a likelihood ratio test. Our evaluation is limited to the case of a known frequency. To examine the small- and large-sample behavior of the estimation procedure for many types of time-dependent arrival processes encountered in previous work, we performed 100 independent replications of twelve selected Poisson processes over two observation intervals. On each replication of each case, we applied the fitting procedure to estimate the parameters of the target process; and we computed the corresponding estimates of the rate and mean-value functions together over the observation interval. Evaluation of the fitting procedure was based on plotted tolerance bands for the rate and mean-value functions with summary statistics for the maximum and average absolute estimation errors in these functions over the observation interval. The experimental results provide substantial evidence of the procedure's numerical stability and practical usefulness.