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–Richter curve. The Monte Carlo tests are also applied to an estimator, b KS , which minimizes the Kolmogorov–Smirnov 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 × 10 5 , 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. Copyright Springer Science+Business Media B.V. 2013