## 4 Application
### Distribution of the Statistically Independent Earthquake Interevent Times
These results are also confirmed by the obtained values of both Information Criteria (AIC and BIC; Tables 3 and 4). In all 10 cases, the Gamma distribution is the one that reports the minimum values for both criteria. Summarizing the results of the comparison of the distributions, although both the Gamma and Weibull distributions can be accepted according to the A-D test in all cases (also the Lognormal one in the data sampled of sub-areas 2 and 8), the \(p\)-values of the test are smaller for the Gamma distribution. The values of AIC and BIC further confirm that Gamma distribution best fits the data since the relevant values are the lowest among the four. By combining the results of the A-D test with those of the information criteria, we found that the Gamma distribution fits better than the other three distributions for all data sets. This result is in agreement with [PERSON]'s analysis [7,8], where the earthquake \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sub-Area** & **Distribution** & **Parameters** & **-Log\(L\)** & **AIC** & **BIC** & **A–D Test** \\ \hline \multirow{8}{*}{9} & \multirow{4}{*}{Gamma} & \(k=0.25\) & & & & **\(p\)-Value** \\  & & [0.21, 0.31] & & & & **\(p\)-Value** \\  & & \(\theta=589.20\) & & & & & \\  & & [386.32, 898.62] & & & & & \\ \cline{2-6}  & \multirow{4}{*}{Lognormal} & \(\mu=2.22\) & & & & & \\  & & [1.46, 2.98] & & 559.08 & 1122.17 & 1127.59 & 0.001 \\  & & \(\sigma=4.05\) & & & & & \\  & & [3.58, 4.66] & & & & & \\ \cline{2-6}  & \multirow{4}{*}{Weibull} & \(\alpha=56.14\) & & & & & \\  & & [32.47, 97.08] & & 543.85 & 1091.70 & 1097.12 & 0.011 \\  & & \(b=0.38\) & & & & & \\  & & [0.30, 0.42] & & & & & \\ \cline{2-6}  & \multirow{2}{*}{Exponential} & \(\mu=149.83\) & & & & & \\  & & [125.43, 182.13] & & 667.05 & 1336.10 & 1338.81 & \(5.405\times 10^{-6}\) \\ \hline \multirow{8}{*}{10} & \multirow{4}{*}{Gamma} & \(k=0.34\) & & & & \\  & & [0.26, 0.45] & & & & \\  & & \(\theta=638.99\) & & 397.61 & 799.12 & 803.68 & 0.419 \\  & & [394.00, 1035.81] & & & & & \\ \cline{2-6}  & \multirow{4}{*}{Lognormal} & \(\mu=3.42\) & & & & & \\  & & [2.64,4.20] & & 414.74 & 833.47 & 837.94 & 0.012 \\  & & [2.78, 3.90] & & & & \\ \cline{2-6}  & \multirow{4}{*}{Weibull} & \(\alpha=122.94\) & & & & & \\  & & [72.54, 208.36] & & 402.44 & 808.88 & 813.35 & 0.127 \\ \cline{2-6}  & & [0.38, 0.57] & & & & \\ \hline \multirow{2}{*}{Exponential} & \(\mu=219.10\) & & & & & & \\  & [157.37, 281.60] & & 440.87 & 883.75 & 885.99 & \(8.695\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._interevent times follow a universal scaling law independent of the region and the \(M_{c}\), and can be modeled by the Gamma distribution. As already stated, the estimated value of shape parameter, \(k\), of Gamma distribution is found to be smaller than one in all the 10 \(\Delta T\) data samples, independent of the temporal  
Figure 8: Comparison of empirical cdf (black lines) and the cdfs of Gamma (red lines), Lognormal (yellow lines), Weibull (blue lines) and Exponential (green lines) applied distributions for sub-areas 6 to 10 (subplots **a–e**). Figure 7: Comparison of empirical cdf (black lines) and the cdfs of Gamma (red lines), Lognormal (yellow lines), Weibull (blue lines) and Exponential (green lines) applied distributions for sub-areas 1 to 5 (subplots **a–e**). crossover magnitude threshold. Specifically, \(k\) ranges from 0.22 to 0.54 in the 10 sub-areas. This is an interesting finding from a seismological point of view because \(k\) plays an important role in earthquake occurrence applications; it could be considered the regulation parameter of the earthquake occurrence process [45]. Specifically, if \(k<1\), the temporal behavior of seismicity could be considered clustered, whereas if \(k>1\), the process tends to be quasi-periodic. In the case of \(k=1\), the process is neither periodic nor clustered, representing the memory-less Poisson process.