Palm distributions of superposed point processes for statistical inference

Abstract: Palm distributions play a central role in the study of point processes and their associated summary statistics. In this paper, we characterize the Palm distributions of the superposition of two independent point processes, establishing a simple mixture representation depending on the point processes' Palm distributions and moment measures. We explore two statistical applications enabled by our main result. First, we consider minimum contrast estimation for superposed point processes based on Ripley's $K$ function. Second, we focus on the class of shot-noise Cox processes and obtain a tractable expression for the Janossy density which leads to maximum likelihood estimation via a novel expectation-maximization algorithm. Both approaches are validated through numerical simulations. Extensions to the superposition of multiple point processes, and higher-order Palm distributions, are also discussed.