Robust estimation of magnitude–frequency relationship parameters

The precise estimation of the a and b parameters of Richter’s magnitude–frequency relationship is of primary importance, since the evaluation of seismicity and assessment of seismic hazard depend on these two parameters. In the literature two popular methods of estimation are available for the estimation of these parameters, namely: least squares and maximum likelihood. However, in implementing these statistical methods, engineers very seldom check the validity of the underlying assumptions with respect to the available data and this may lead to serious problems. Under non-normality least squares estimators (LSEs) are neither efficient nor robust and maximum likelihood estimators (MLEs) are elusive due to numerous complexities. A robust estimation procedure, the modified maximum likelihood method (MML), can be utilized to estimate the unknown parameters a and b in such situations. The resulting estimators are explicit functions of sample observations and are shown to be considerably more efficient than the commonly used least squares estimators. In addition, we demonstrate that the MML estimators are more appropriate to estimate the parameters of Richter’s magnitude–frequency relationship based on the comparison of their performance with those of the least squares estimators by using the seismic database on earthquakes recorded in Turkey.