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Official Journal of the Asia Oceania Geosciences Society (AOGS)

Table 1 Probability distributions and results of statistical inference of natural times

From: Nowcasting earthquakes in Sulawesi Island, Indonesia

Distribution Density function \(\left( {t > 0} \right)\) Statistical inference
MLE AIC K–S
Exponential (best-fit) \(\frac{1}{\alpha }e^{{ - \frac{t}{\alpha }}}\) \(\hat{\alpha } =\) 191.9540 1090.7625 0.0567
Gamma \(\frac{1}{\Gamma \left( \beta \right)}\frac{{t^{\beta - 1} }}{{\alpha^{\beta } }}e^{{ - \frac{t}{\alpha }}}\) \(\hat{\alpha } =\) 231.1659
\(\hat{\beta } =\) 0.8304
1093.8716 0.0888
Weibull \(\frac{\beta }{{\alpha^{\beta } }}t^{\beta - 1} e^{{ - \left( {\frac{t}{\alpha }} \right)^{\beta } }}\) \(\hat{\alpha } =\) 191.0590
\(\hat{\beta } =\) 0.9888
1092.7447 0.0596
Exponentiated exponential \(\alpha \beta \left( {1 - e^{ - \alpha t\,} } \right)^{\beta - 1} \,e^{ - \alpha t\,}\) \(\hat{\alpha } =\) 197.3369
\(\hat{\beta } =\) 0.9580
1092.6662 0.0617
  1. MLE maximum likelihood estimation, AIC Akaike information criterion, K–S Kolmogorov–Smirnov