Method of Moments Estimators and Multu-step MLE for Poisson Processes

TL;DR

This paper introduces method of moments and multi-step maximum likelihood estimators for inhomogeneous Poisson processes, demonstrating their consistency, asymptotic normality, and efficiency with a focus on computational simplicity.

Contribution

It presents a novel two-step MLE construction method that is computationally simple and proves the estimators' statistical properties for inhomogeneous Poisson processes.

Findings

Method of moments estimators are consistent and asymptotically normal.

Multi-step MLEs are consistent and asymptotically efficient.

The proposed approach simplifies computation of estimators.

Abstract

We introduce two types of estimators of the finite-dimensional parameters in the case of observations of inhomogeneous Poisson processes. These are the estimators of the method of moments and multi-step MLE. It is shown that the estimators of the method of moments are consistent and asymptotically normal and the multi-step MLE are consistent and asymptotically efficient. The construction of multi-step MLE-process is done in two steps. First we construct a consistent estimator by the observations on some learning interval and then this estimator is used for construction of one-step and two-step MLEs. The main advantage of the proposed approach is its computational simplicity.