Bootstrap determination of the reliability of b-values: an assessment of statistical estimators with synthetic magnitude series

We consider some practical issues of the determination of the b-value of sequences of magnitudes with the bootstrap method for short series of length L and various quantization levels $\Updelta m$ of the magnitude. Preliminary Monte Carlo tests performed with $\Updelta m = 0$ demonstrate the superiority of the maximum likelihood estimator b _MLE, and the inconsistency of the, yet often used, b _LR estimator defined as the least-squares slope of the experimental Gutenberg?CRichter curve. The Monte Carlo tests are also applied to an estimator, b _KS, which minimizes the Kolmogorov?CSmirnov distance between the cumulative distribution of magnitudes and a power-law model. Monte Carlo tests of discrete versions of the b _MLE and b _KS estimators are done for $\Updelta m = \{0.1, 0.2, 0.3 \}$ and used as reference to evaluate the performance of the bootstrap determination of b. We show that all estimators provide b estimates within 10?% error for L????100 and if a large number, n?=?2?×?10⁵, of bootstrapped sample series is used. A resolution test done with $\Updelta m = 0.1$ reveals that a clear distinction between b?=?0.8, 1.0, and 1.2 is obtained if L????200.