Contrast estimation for parametric stationary determinantal point processes

TL;DR

This paper develops and proves the consistency and asymptotic normality of minimum contrast estimators for parametric stationary determinantal point processes, which model repulsive spatial patterns, using Ripley's K-function and pair correlation function.

Contribution

It introduces new statistical estimation methods for determinantal point processes and establishes their theoretical properties under general conditions.

Findings

Proves strong consistency of the estimators.

Establishes asymptotic normality of the estimators.

Utilizes the Brillinger mixing property for proofs.

Abstract

We study minimum contrast estimation for parametric stationary determi-nantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K-function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation).