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Table 2 Probability distributions, density functions, and associated Fisher Information Matrices (FIM)

From: Earthquake interevent time distribution in Kachchh, Northwestern India

  Density function Parameters FIM (I(θ))
Distribution PDF Domain Role Domain  
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 − 1e − 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
  1. a ψ(x) and ψ′(x) denote the digamma function and its first derivate
  2. bThe FIMs of exponentiated Rayleigh and exponentiated Weibull distributions are not calculated, as the FIM of exponentiated Rayleigh distribution contains highly non-linear implicit terms (Kundu and Raqab 2005), and the FIM of exponentiated Weibull distribution is not completely known (Pal et al. 2006)