How large does chi-squared get by chance? A measured null distribution for the combination step of Ellis et al.

Abstract

Ellis et al. [Phys. Rev. D 99, 083009 (2019)] combine the K-reduced distributions of eight Fermi-LAT gamma-ray bursts into a joint bound on the scale of linear Lorentz violation. The combination yields a raw chi-squared of 261, from which they conclude that the sources are not identical. That conclusion presupposes that a chi-squared of this sizewould be rare under the null hypothesis. This work tests the presupposition by measuring the null distribution of chi-squared itself: photon energies are permuted among fixed arrival times, the estimator is re-evaluated on every permutation, and the eight results are combined by the same prescriptions — 200,001 times per burst and time window, across a family of forty time windows. The estimator is evaluated without a grid: the weighted moments are polynomials in the compensation parameter, so the maximum of the kurtosis is a root of a fifth-degree polynomial and is obtained exactly. The observed value chi-squared = 267.05 corresponds to p values between 0.351 and 1.000; in none of the twenty-eight assessable windows is it rare. The combination step does not support the conclusion drawn from it. As secondary results the origin of the published number 261 is identified — it follows from a redshift for GRB 160509A given as z = 1.60 in Table Iof that work, while its own figures compute with z = 1.17 — and the estimator is shown to possess a second, distant maximum which wins for certain data selections. This work does not refute the published bound; it shows that one of its intermediate steps establishes nothing. Deposited in German and English, plus the version submitted to Physical Review D (REVTeX 4.2). Analysis code and raw results: doi:10.5281/zenodo.22541521Preregistrations: doi:10.5281/zenodo.22285126,doi:10.5281/zenodo.22287439, doi:10.5281/zenodo.22288065