Asymptotic Goodness-of-Fit Tests for Point Processes Based on Scaled Empirical K-Functions

TL;DR

This paper develops asymptotic goodness-of-fit tests for point processes using scaled empirical K-functions, enabling hypothesis testing based on observed data and theoretical models without requiring explicit knowledge of intensities.

Contribution

It introduces new asymptotic distributions for scaled empirical K-functions, facilitating goodness-of-fit tests and comparisons between independent point processes.

Findings

Normalized differences converge to Gaussian processes

Discrepancy measures with known limit distributions are proposed

Tests do not require explicit intensity or K-function knowledge

Abstract

We study sequences of scaled edge-corrected empirical (generalized) K-functions (modifying Ripley's K-function) each of them constructed from a single observation of a $d$-dimensional fourth-order stationary point process in a sampling window W_n which grows together with some scaling rate unboundedly as n --> infty. Under some natural assumptions it is shown that the normalized difference between scaled empirical and scaled theoretical K-function converges weakly to a mean zero Gaussian process with simple covariance function. This result suggests discrepancy measures between empirical and theoretical K-function with known limit distribution which allow to perform goodness-of-fit tests for checking a hypothesized point process based only on its intensity and (generalized) K-function. Similar test statistics are derived for testing the hypothesis that two independent point processes in W_n have the same distribution without explicit knowledge of their intensities and K-functions.