From: Earthquake interevent time distribution in Kachchh, Northwestern India
 | Density function | Parameters | FIM (I(θ)) | ||
---|---|---|---|---|---|
Distribution | Domain | Role | Domain | Â | |
Exponential | \( \frac{1}{\alpha }{e}^{-\frac{t}{\alpha }} \) | t > 0 | α −scale | α > 0 | \( \frac{1}{\alpha^2} \) |
Gammaa | \( \frac{1}{\varGamma \left(\beta \right)}\frac{t^{\beta -1}}{\alpha^{\beta }}{e}^{-\frac{t}{\alpha }} \) | t > 0 | α −scale | \( \begin{array}{l}\alpha >0\\ {}\beta >0\;\end{array} \) | \( \left[\begin{array}{cc}\hfill \frac{\beta }{\alpha^2}\hfill & \hfill \frac{1}{\alpha}\hfill \\ {}\hfill \frac{1}{\alpha}\hfill & \hfill {\psi}^{\prime}\left(\beta \right)\hfill \end{array}\right] \) |
β −shape | |||||
Lognormal | \( \frac{1}{t\beta \sqrt{2\pi }} \exp \left[-\frac{1}{2}{\left(\frac{ \ln t-\alpha }{\beta}\right)}^2\right] \) | t > 0 | α −log-scale | \( \begin{array}{l}-\infty <\alpha <\infty \\ {}\beta >0\end{array} \) | \( \left[\begin{array}{cc}\hfill \frac{2}{\beta^2}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{1}{\beta^2}\hfill \end{array}\right] \) |
β −shape | |||||
Weibulla | \( \frac{\beta }{\alpha^{\beta }}{t}^{\beta -1}{e}^{-{\left(\frac{t}{\alpha}\right)}^{\beta }} \) | t > 0 | α −scale | \( \begin{array}{l}\alpha >0\\ {}\beta >0\;\end{array} \) | \( \left[\begin{array}{cc}\hfill \frac{\beta^2}{\alpha^2}\hfill & \hfill -\frac{1}{\alpha}\left(1+\psi (1)\right)\hfill \\ {}\hfill -\frac{1}{\alpha}\left(1+\psi (1)\right)\hfill & \hfill \frac{1}{\beta^2}\left({\psi}^{\prime }(1)+{\psi}^2(2)\right)\hfill \end{array}\right] \) |
β −shape | |||||
Levy | \( \sqrt{\frac{\alpha }{2\pi }}\frac{{\operatorname{e}}^{-\frac{\alpha }{2t}}\;}{t^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} \) | t > 0 | α −scale | α > 0 | \( \frac{1}{2{\alpha}^2} \) |
Maxwell | \( \sqrt{\frac{2}{\pi }}\frac{t^2}{\alpha^3} \exp \left[-\frac{1}{2}{\left(\frac{t}{\alpha}\right)}^2\right] \) | t > 0 | α −scale | α > 0 | \( \frac{6}{\alpha^2} \) |
Pareto | \( \beta \frac{\alpha^{\beta }}{t^{\beta +1}} \) | t > α | α −scale | \( \begin{array}{l}\;t>\alpha >0\\ {}\beta >0\end{array} \) | \( \left[\begin{array}{cc}\hfill \frac{\beta }{\alpha^2\left(\beta +2\right)}\hfill & \hfill -\frac{1}{\alpha \left(\beta +1\right)}\hfill \\ {}\hfill -\frac{1}{\alpha \left(\beta +1\right)}\hfill & \hfill \frac{1}{\beta^2}\hfill \end{array}\right] \) |
β −shape | |||||
Rayleigh | \( \frac{t}{\alpha^2} \exp \left(-\frac{t^2}{2{\alpha}^2}\right) \) | t > 0 | α −scale | α > 0 | \( \frac{4}{\alpha^2} \) |
Inverse Gaussian (Brownian Passage Time) | \( \sqrt{\frac{\beta }{2\pi {t}^3}} \exp \left[-\frac{\beta {\left(t-\alpha \right)}^2}{2{\alpha}^2t}\right] \) | t > 0 | β/α −shape | \( \begin{array}{l}\alpha >0\\ {}\beta >0\;\end{array} \) | \( \left[\begin{array}{cc}\hfill \frac{1}{2{\beta}^2}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{\beta }{\alpha^3}\hfill \end{array}\right] \) |
Inverse Weibull (Frechet)a | \( \beta {\alpha}^{\beta }{t}^{-\beta -1}{e}^{-{\left(\frac{t}{\alpha}\right)}^{-\beta }} \) | t > 0 | α −scale | \( \begin{array}{l}\alpha >0\\ {}\beta >0\;\end{array} \) | \( \left[\begin{array}{cc}\hfill \frac{\beta^2}{\alpha^2}\hfill & \hfill \frac{1}{\alpha}\left(1+\psi (1)\right)\hfill \\ {}\hfill \frac{1}{\alpha}\left(1+\psi (1)\right)\hfill & \hfill \frac{1}{\beta^2}\left({\psi}^{\prime }(1)+{\psi}^2(2)\right)\hfill \end{array}\right] \) |
β −shape | |||||
Exponentiated exponentiala | αβ(1 − e − αt )β − 1  e − αt | t > 0 | 1/α −scale | \( \begin{array}{l}\alpha >0\\ {}\beta >0\;\end{array} \) | \( \left[\begin{array}{cc}\hfill {a}_{11}\hfill & \hfill {a}_{12}\hfill \\ {}\hfill {a}_{21}\hfill & \hfill {a}_{22}\hfill \end{array}\right] \) \( \begin{array}{l}{a}_{11}=\frac{1}{\alpha^2}\left[1+\frac{\beta \left(\beta -1\right)}{\beta -2}\left({\psi}^{\prime }(1)-{\psi}^{\prime}\left(\beta -1\right)\right)+{\left(\psi \left(\beta -1\right)-\psi (1)\right)}^2\right]\\ {}\kern1.32em -\frac{\beta }{\alpha^2}\left[{\psi}^{\prime }(1)-\psi \left(\beta \right)+{\left(\psi \left(\beta \right)-\psi (1)\right)}^2\right];\kern0.24em \beta \ne 2\\ {}{a}_{12}={a}_{21}=\frac{1}{\alpha}\left[\frac{\beta }{\beta -1}\left(\psi \left(\beta \right)-\psi (1)\right)-\left(\psi \left(\beta +1\right)-\psi (1)\right)\right];\kern0.24em \beta \ne 1\\ {}{a}_{22}=\frac{1}{\beta^2}\end{array} \) |
β −shape | |||||
Exponentiated Rayleigh (Burr Type X)b | \( \frac{2\beta t}{\alpha^2}{e}^{-{\left(\frac{t}{\alpha}\right)}^2}{\left(1-{e}^{-{\left(\frac{t}{\alpha}\right)}^2}\right)}^{\beta -1} \) | t > 0 | α −scale | \( \begin{array}{l}\alpha >0\\ {}\beta >0\;\end{array} \) | − |
β −shape | |||||
Exponentiated Weibullb | \( \frac{\beta \gamma }{\alpha }{\left(\frac{t}{\alpha}\right)}^{\beta -1}{e}^{-{\left(\frac{t}{\alpha}\right)}^{\beta }}{\left(1-{e}^{-{\left(\frac{t}{\alpha}\right)}^{\beta }}\right)}^{\gamma -1} \) | t > 0 | α −scale | \( \begin{array}{l}\alpha >0\\ {}\beta >0\;\\ {}\gamma >0\;\end{array} \) | − |
β −shape | |||||
γ −shape |