## 4 Application
### Distribution of the Statistically Independent Earthquake Interevent Times
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Sub-Area** & **Distribution** & **Parameters** & **-Log\(L\)** & **AIC** & **BIC** &
\begin{tabular}{c} **A–D Test** \\ _p_**-Value** \\ \end{tabular} \\ \hline \multirow{6}{*}{6} & \multirow{3}{*}{Gamma} & \(k\) = 0.54 & \multirow{3}{*}{596.96} & \multirow{3}{*}{1197.91} & \multirow{3}{*}{1203.14} & \multirow{3}{*}{0.489} \\  & & \(\theta\) = 297.33 & & & & \\  & & [209.32, 422.38] & & & & \\ \cline{2-7}  & \multirow{3}{*}{Lognormal} & \(\mu\) = 3.91 & \multirow{3}{*}{626.13} & \multirow{3}{*}{1256.25} & \multirow{3}{*}{1261.48} & \multirow{3}{*}{0.001} \\  & & \(\sigma\) = 2.40 & & & & \\  & & [2.11, 2.78] & & & & \\ \cline{2-7}  & \multirow{3}{*}{Weibull} & \(\begin{array}{c}a=129.01\\ [95.63, 174.04] \\ [209.32, 422.38] \\ \end{array}\) & \(600.18\) & \multirow{3}{*}{1204.36} & \multirow{3}{*}{1209.59} & \multirow{3}{*}{0.261} \\  & & \(\mu\) = 160.18 & \multirow{3}{*}{613.71} & \multirow{3}{*}{1229.41} & \multirow{3}{*}{1232.03} & \multirow{3}{*}{0.004} \\  & & \(\sigma\) = 102.9 & \multirow{3}{*}{320.67} & \multirow{3}{*}{645.33} & \multirow{3}{*}{649.42} & \multirow{3}{*}{0.628} \\  & & \(\theta\) = 1092.53 & & & & \\  & & [614.10, 1943.69] & & & & \\ \hline \multirow{6}{*}{7} & \multirow{3}{*}{Lognormal} & \(\mu\) = 3.09 & \multirow{3}{*}{333.77} & \multirow{3}{*}{671.55} & \multirow{3}{*}{675.63} & \multirow{3}{*}{0.034} \\  & & \(\sigma\) = 3.86 & & & & \\  & & [3.261, 4.73] & & & & \\ \hline \multirow{6}{*}{Weibull} & \(\begin{array}{c}a=118.51\\ [58.18, 241.43] \\ [0.31, 0.47] \\ \end{array}\) & \(324.89\) & \multirow{3}{*}{653.78} & \multirow{3}{*}{657.87} & \multirow{3}{*}{0.171} \\  & & \(\theta\) = 295.46 & & & & \\  & & [231.59, 390.10] & & & & \\ \hline \multirow{6}{*}{Gamma} & \(k\) = 0.26 & \multirow{6}{*}{252.93} & \multirow{6}{*}{509.86} & \multirow{6}{*}{513.43} & \multirow{6}{*}{0.315} \\  & & \(\theta\) = 1452.99 & & & & \\  & & [745.40, 2832.30] & & & & \\ \hline \multirow{6}{*}{Lognormal} & \(\mu\) = 3.18 & \multirow{6}{*}{259.64} & \multirow{6}{*}{523.28} & \multirow{6}{*}{526.85} & \multirow{6}{*}{0.189} \\  & & \(\sigma\) = 3.73 & & & & \\  & & [3.08, 4.72] & & & & \\ \hline \multirow{6}{*}{Weibull} & \(\begin{array}{c}\alpha=130.99\\ [55.36, 309.98] \\ [0.28, 0.46] \\ \end{array}\) & \(255.63\) & \multirow{6}{*}{515.26} & \multirow{6}{*}{518.83} & \multirow{6}{*}{0.215} \\  & & \(\theta\) = 0.36 & & & & \\ \hline \multirow{6}{*}{Exponential} & \(\mu\) = 373.98 & \multirow{6}{*}{304.66} & \multirow{6}{*}{611.33} & \multirow{6}{*}{613.11} & \multirow{6}{*}{1.363} \times 10^{-5}\) \\  & & [284.09,514.69] & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: 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 6 to 10. The Anderson-Darling (A-D) test is then applied to each sample in order to compare the distributions derived via the MLE parameter estimates and empirical cdf (Figures 7 and 8 for sub-areas 1-5 and 6-10, respectively). Results of the A-D goodness of fit test (Tables 3 and 4) show that in all cases the Exponential distribution can be rejected since the respective \(p\)-values are getting lower than the 0.05 level of significance (\(p\)-value < 0.05). Similarly, the Lognormal distribution is also rejected in 8 out of 10 cases. The rejected distributions report \(p\)-values larger than 0.05 for the central Ionian Islands and central Greece areas (sub-areas 2 and 8; Table 3). The Gamma and Weibull distributions perform better than the others in all the 10 \(\Delta T\) data sets. Gamma distribution exhibits slightly better performance than the Weibull since the respective \(p\)-values are always larger than the ones of Weibull.