Takacs-Fiksel estimation as a special case of Point Process Learning

Journal article, 2026

In the context of parameter estimation for Gibbs point processes, the state-of-the-art method is Takacs-Fiksel estimation, of which pseudolikelihood estimation is a special case. An alternative method is the recently proposed Point Process Learning approach, based on point process cross-validation and point process prediction errors. Since both Takacs-Fiksel estimation and Point Process Learning are motivated by the Georgii–Nguyen–Zessin formula, which defines Gibbs point processes, in this paper we study Point Process Learning in relation to Takacs-Fiksel estimation. We show that, upon applying appropriate scaling and letting the cross-validation regime tend to leave-one-out cross-validation in Point Process Learning, averages of prediction errors converge to the innovation-based loss function in Takacs-Fiksel estimation. We further provide an empirical risk formulation of Point Process Learning, which highlights the nature of our asymptotic results, and show that the underlying convergence mechanism can be partially understood through a conditional law of large numbers for statistics of conditionally independent thinnings. We finally illustrate our theoretical findings through simulations for a Strauss process, focusing on both convergence diagnostics and comparison of parameter estimation performance between the two approaches.