Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes

TLDR: In this paper, the authors extended the maximum likelihood estimation of the earthquake parameters N o and β in the relation N = N o exp (− βm ) to the case of events grouped in magnitude with each group observed over individual time periods.

Abstract: Maximum likelihood estimation of the earthquake parameters N o and β in the relation N = N o exp (− βm ) is extended to the case of events grouped in magnitude with each group observed over individual time periods. Asymptotic forms of the equation for β reduce to the estimators given for different special cases by Aki (1965), Utsu (1965, 1966), and Page (1968). The estimates of β are only approximately chi-square distributed. For sufficiently large numbers of events, they can be estimated from the curvature of the log-likelihood function. Sample calculations for three earthquake source zones in western Canada indicate that for well-constrained data sets, the often-used, least-squares estimation procedures lead to compatible results, but for less well-defined data sets, the effect of subjective plotting and weighting methods used for least-squares fitting leads to appreciably different parameters.