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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 − 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

  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)