A technical note on the bias in the estimation of the b-value and its uncertainty through the least squares technique

We investigate conceptually, analytically, and numerically the biases in the estimation of the b-value of the Gutenberg-Richter Law and of its uncertainty made through the least squares technique. The biases are introduced by the cumulation operation for the cumulative form of the Gutenberg-Richter Law, by the logarithmic transformation, and by the measurement errors on the magnitude. We find that the least squares technique, applied to the cumulative and binned form of the Gutenberg-Richter Law, produces strong bias in the b-value and its uncertainty, whose amplitudes depend on the size of the sample. Furthermore, the logarithmic transformation produces two different endenidc bends in the Log(N) versus M curve. This means that this plot might produce fake significant departures from the Gutenberg-Richter Law. The effect of the measurement errors is negligible compared to those of cumulation operation and logarithmic transformation. The results obtained show that the least squares technique should never be used to determine the slope of the Gutenberg-Richter Law and its uncertainty.