## 4 Application
### Distribution of the Statistically Independent Earthquake Interevent Times  
After the detection of the magnitude thresholds above where the \(\Delta T\) intervals are considered statistically independent, the statistical distribution of those data sets is investigated, with the application of the Weibull, Lognormal, Gamma and Exponential distributions. The parameters of each distribution are estimated via the MLE method according to their respective formulations, along with their 95% confidence intervals. Tables 3 and 4 present the estimation results, along with the respective 95% confidence intervals and the values of the negative log-likelihood functions for sub-areas 1-5 and 6-10, respectively. An interesting point arising from the MLE approach is that the shape parameters of both the Gamma and Weibull distributions, \(k\) and \(b\), respectively, obtain values below one in all cases (in all 10 sub-areas). This is an indicator of a clustering-type behavior of earthquake occurrence [45]. \begin{table}
\begin{tabular}{c c c c} \hline
**Sub-Area** & \(M_{thr}\)**Identified Using the** & \(M_{thr}\)**Identified Using the** & **Sample Size** \\  & _ACF_ & **PACF Values** & **Ljung–Box Q-Test** & **(Interevent Times)** \\ \hline
1-Western Greece and Albania & 5.1 & 5.1 & 50 \\
2-Central Ionian Islands & 4.9 & 4.9 & 88 \\
3-Western Hellenic Arc & 4.8 & 4.8 & 260 \\
4-Eastern Hellenic Arc & 5.0 & 5.0 & 130 \\
5-Southeast Aegean Sea & 5.0 & 5.0 & 80 \\
6-Southwest Aegean Sea and Peloponnese & 4.7 & 4.7 & 100 \\
7-Corinth Gulf & 4.9 & 4.9 & 57 \\
8-Central Greece & 5.0 & 5.0 & 45 \\
9-North Aegean Sea & 5.0 & 5.0 & 111 \\
10-Northern Greece and Balkans & 4.8 & 4.8 & 69 \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of the identification of the magnitude threshold (\(M_{thr}\)) above which interevent time data, \(\Delta T\), are statistically independent for the 10 sub-areas of the broader Aegean region. Figure 6: Exponent \(\alpha\) of Detrended Fluctuations Analysis as a function of \(M_{thr}\) for the 10 sub-areas (subplots **a-j**). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Sub-Area** & **Distribution** & **Parameters** & **-LogL** & **AIC** & **BIC** & **A-D Test** \\ \hline \multirow{8}{*}{1} & \multirow{4}{*}{Gamma} & \(k=0.38\) & \multirow{4}{*}{310.25} & \multirow{4}{*}{624.50} & \multirow{4}{*}{628.28} & \multirow{4}{*}{0.690} \\  & & & [0.29, 0.53] & & & & \\  & & \(\theta=864.34\) & & & & & \\  & & [497.05, 1503.01] & & & & & \\ \cline{2-7}  & \multirow{4}{*}{Lognormal} & \(\mu=4.08\) & \multirow{4}{*}{318.72} & \multirow{4}{*}{641.43} & \multirow{4}{*}{645.22} & \multirow{4}{*}{0.049} \\  & & & [3.28, 4.87] & & & & \\  & & \(\sigma=2.76\) & & & & & \\  & & [2.30, 3.45] & & & & & \\ \cline{2-7}  & \multirow{4}{*}{Weibull} & \(\alpha=192.99\) & \multirow{4}{*}{310.99} & \multirow{4}{*}{625.99} & \multirow{4}{*}{629.77} & \multirow{4}{*}{0.445} \\  & & & \(b=0.52\) & & & & \\  & & [0.42, 0.65] & & & & & \\ \cline{2-7}  & \multirow{4}{*}{Exponential} & \(\mu=331.79\) & \multirow{4}{*}{333.42} & \multirow{4}{*}{668.84} & \multirow{4}{*}{670.73} & \multirow{4}{*}{2.531 \(\times\) 10\({}^{-5}\)} \\  & & & [255.46, 448.48] & & & & \\ \cline{2-7}  & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MLE parameters estimates, 95% confidence intervals, log-likelihood values calculation, Akaike and Bayesian Information Criteria calculated values (AIC and BIC, respectively), along with the estimated \(p\)-values of the Anderson–Darling Goodness of Fit test for the four applied statistical distributions on the statistically independent interevent times data, \(\Delta T\), of the sub-areas 1 to 5.