The M(agnitude)-test

The M(agnitude)-test measures the consistency of a forecast with the observed distribution of earthquake magnitudes, disregarding the rate and spatial distribution specified by the forecast. The joint log-likelihood of the observed catalog is computed conditional on a magnitude-normalized forecast, and this value is compared with the expected distribution of joint log-likelihoods assuming that the magnitude-normalized forecast is correct. If the joint log-likelihood of the observed catalog is exceedingly low relative to the expected distribution, this indicates that the forecast has not predicted well the magnitude distribution of earthquakes.