Numerical Realization of Minimum Distance Estimation for Poisson Processes

Abstract The statistical inference for minimum distance parametric estimation has been well developed in recent years, concerning the asymptotic behaviors of the minimum L 1 -distance estimator, minimum L 2 -distance estimator and minimum Skorohod-distance estimator. The question is how to realize the numerical solution of the minimum Skorohod-distance estimator, due to its complicated analytic structure. This paper aims to discuss the minimum distance estimation problem for Poisson processes. The main contribution includes the consistency and the approximate numerical solution of the minimum Skorohod-distance estimator, accompanied with the study of the minimum L 1 -distance estimator and minimum L 2 -distance estimator. In the numerical examples, three estimators exhibit the consistency trend. The mean square error of the minimum L 1 -distance estimator and minimum L 2 -distance estimator are comparable to, or superior to the minimum Skorohod-distance estimator in the regular cases, while the results are reversed in the change-point case. The bias of the minimum Skorohod-distance estimator is either superior or comparable to the others in the regular cases.