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\begin{document}

\title{b-more-incomplete and b-more-positive: Insights on A Robust Estimator of
Magnitude Distribution}



\author[1]{Eugenio Lippiello}%
\author[2]{Giuseppe Petrillo}%
\affil[1]{University of Campania}%
\affil[2]{The Institute of Statistical Mathematics}%


\vspace{-1em}



  
  \date{May 18, 2024}


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\begin{abstract}
The \$b\$-value in earthquake magnitude-frequency distribution
quantifies the relative frequency of large versus small earthquakes.
Monitoring its evolution could provide fundamental insights into
temporal variations of stress on different fault patches. However,
genuine \$b\$-value changes are often difficult to distinguish from
artificial ones induced by temporal variations of the detection
threshold.\\
A highly innovative and effective solution to this issue has recently
been proposed by van der Elst (2021) by means of the b-positive
estimator, which is based on analyzing only the positive differences in
magnitude between successive earthquakes.\\
Here, we demonstrate the robustness of the estimator, which remains
largely unaffected by detection issues due to the properties of
conditional probability. We illustrate that this robustness can be
further improved by considering positive differences in magnitude, not
only between successive earthquakes but also between different pairs of
earthquakes. This generalized approach, defined as the ``b-more-positive
estimator,'' enhances efficiency by providing a precise estimate of the
\$b\$-value while including a larger number of earthquakes from an
incomplete catalog. However, our analysis reveals that the accuracy of
the \$b\$ estimators diminishes when earthquakes below the completeness
threshold are included in the catalog. This leads to the paradoxical
observation that greater efficiency is achieved when the catalog is more
incomplete. To address this, we introduce the ``b-more-incomplete
estimator'', where the b-more-positive estimator is applied only after
artificially filtering the instrumental catalog to make it more
incomplete. Our findings show the superior efficiency of the
b-more-incomplete method.%
\end{abstract}%



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