Asymptotic approximation of the likelihood of stationary determinantal point processes

TL;DR

This paper introduces a new asymptotic likelihood approximation for stationary determinantal point processes that is more versatile and computationally efficient than previous methods, with theoretical support and practical validation.

Contribution

It proposes a novel asymptotic likelihood approximation for stationary DPPs that overcomes previous limitations and provides explicit variance estimation, supported by theoretical analysis.

Findings

The new approximation is applicable to non-rectangular windows.

It is faster and requires no tuning parameters.

Simulation results show improved performance over existing methods.

Abstract

Continuous determinantal point processes (DPPs) are a class of repulsive point processes on $\mathbb{R}^d$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards $\mathbb{R}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on $\mathbb{R}^d$ and compare favourably to common alternative estimation methods for continuous DPPs.