Minimax estimation of the structure factor of spatial point processes

TL;DR

This paper develops a minimax optimal estimator for the structure factor of stationary spatial point processes, providing theoretical bounds, a practical multitaper method, and data-driven tuning with numerical validation.

Contribution

It introduces a minimax optimal multitaper estimator for the structure factor, with theoretical bounds, a data-driven selection procedure, and empirical validation.

Findings

Achieves the optimal rate of convergence in squared risk.

Provides a chi-square-type concentration bound under mixing conditions.

Demonstrates practical effectiveness through numerical experiments.

Abstract

We investigate the problem of estimating the structure factor, or spectra, of stationary spatial point processes. In the first part, we establish a minimax lower bound for this estimation problem, using an approach tailored to second-order properties of spatial point processes. Although not the main focus, this methodology also extends naturally to a minimax lower bound for the estimation of the pair correlation function of spatial point processes. In the second part, we construct a multitaper estimator that achieves the optimal rate of convergence in squared risk. Under a Brillinger-mixing condition, we further establish a chi-square-type concentration bound. Finally, we propose a data-driven procedure for selecting the number of tapers, supported by an oracle inequality, and we demonstrate the practical effectiveness of the method through numerical experiments.