Recursive Estimation of the Intensity Function of the Non-homogeneous Poisson Process

This study focuses on the recursive nonparametric estimation of the intensity function associated with a nonhomogeneous Poisson process. Accurately estimating the intensity function is crucial for understanding the dynamics of events in fields such as finance, neuroscience, and environmental monitoring. While traditional nonparametric estimators are theoretically robust, their reliance on the entire dataset for every update makes them impractical for real-time applications. To overcome this limitation, we introduce a recursive estimator that supports efficient, online updates as new data becomes available. This approach significantly lowers computational overhead while maintaining strong statistical reliability. We thoroughly analyze the asymptotic behavior of the proposed estimator, paying particular attention to the Asymptotic Mean Integrated Squared Error (AMISE), a key measure of estimation accuracy. Additionally, we compare the performance of our recursive estimator with Cucala’s non-recursive method. The results reveal that our approach achieves equivalent or superior accuracy in terms of AMISE, particularly in large-sample scenarios. A computational performance comparison further underscores the advantages of the proposed method, demonstrating its substantial reduction in execution time and its suitability for applications requiring rapid processing.