Generalized Pareto Distribution
Calculate the probability density function, lower cumulative distribution function, and upper cumulative distribution function of the generalized Pareto distribution.
| \(\normalsize Generalized\ Pareto\ distribution\\ (1)\ probability\ density\\ \hspace{30px}f(x,\mu,\sigma,\xi)= \left\{ \begin{array}{l} \large \frac{1}{\sigma} \left\{ 1 + \frac{\xi (x- \mu )}{\sigma} \right\}^{\small \left( - \frac{1}{\xi}-1 \right)} \hspace{30px} {\small \xi \not= 0 } \\ \frac{1}{\sigma} \exp \left\{ - \frac{(x- \mu )}{\sigma} \right\} \hspace{65px} {\small \xi = 0 } \end{array} \right. \\ (2)\ lower\ cumulative\ distribution\\ \hspace{30px}P(x,\mu,\sigma,\xi)= \left\{ \begin{array}{l} \large 1- \left\{ 1 + \frac{\xi (x- \mu )}{\sigma} \right\}^{\small - \frac{1}{\xi}} \hspace{30px} {\small \xi \not= 0 } \\ 1 - \exp \left( - \frac{x- \mu}{\sigma} \right) \hspace{50px} {\small \xi = 0 } \end{array} \right. \\ (3)\ upper\ cumulative\ distribution\\ \hspace{30px}Q(x,\mu,\sigma,\xi)= \left\{ \begin{array}{l} \large \left\{ 1 + \frac{\xi (x- \mu )}{\sigma} \right\}^{\small - \frac{1}{\xi}} \hspace{30px} {\small \xi \not= 0 } \\ \exp \left( - \frac{x- \mu}{\sigma} \right) \hspace{50px} {\small \xi = 0 } \end{array} \right. \\ \) |
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