# Second-order variational equations for spatial point processes with a view to pair correlation function estimation

## TL;DR

This paper develops second-order variational equations for spatial point processes, enabling closed-form parameter estimation for pair correlation functions, and introduces a method to fit orthogonal series expansions of these functions.

## Contribution

It introduces a novel application of second-order variational equations to spatial point processes, facilitating explicit parameter estimation for pair correlation functions.

## Key findings

- Variational equations can be used to derive closed-form estimators.
- Method allows fitting orthogonal series expansions of pair correlation functions.
- Applicable to log linear parametric models for pair correlation functions.

## Abstract

Second-order variational type equations for spatial point processes are established. In case of log linear parametric models for pair correlation functions, it is demonstrated that the variational equations can be applied to construct estimating equations with closed form solutions for the parameter estimates. This result is used to fit orthogonal series expansions of log pair correlation functions of general form.