A functional central limit theorem for the K-function with an estimated intensity function

TL;DR

This paper establishes the asymptotic distribution of the inhomogeneous K-function estimator, accounting for estimated intensity functions, thus advancing the understanding of its statistical properties in spatial point process analysis.

Contribution

It derives the functional central limit theorem for the inhomogeneous K-function estimator with an estimated intensity, overcoming previous limitations in the literature.

Findings

Provides the asymptotic distribution for the estimator

Handles the case of estimated intensity functions

Extends previous results to inhomogeneous processes

Abstract

The $K$-function is arguably the most important functional summary statistic for spatial point processes. It is used extensively for goodness-of-fit testing and in connection with minimum contrast estimation for parametric spatial point process models. It is thus pertinent to understand the asymptotic properties of estimates of the $K$-function. In this paper we derive the functional asymptotic distribution for the $K$-function estimator. Contrary to previous papers on functional convergence we consider the case of an inhomogeneous intensity function. We moreover handle the fact that practical $K$-function estimators rely on plugging in an estimate of the intensity function. This removes two serious limitations of the existing literature.