Official Journal of the Asia Oceania Geosciences Society (AOGS)
Table 1 Probability distributions and results of statistical inference of natural times
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 |