We derive the cosmological relationships between the observed and emitted quantities, and we arrive at a prediction that can be tested on the ensembles of bursts with determined redshifts. This analysis is independent on the assumed cosmology, on the observational biases, as well as on any gamma-ray burst model.
Four different samples are studied: 8 BATSE bursts with redshifts, 13 bursts with derived pseudo-redshifts, 134 Swift bursts with redshifts, and 6 Fermi bursts with redshifts.