Generalized Pareto

Parameters

tail index \(\xi\), llocation parameter \(\mu\), and scale parameter \(\sigma\)

Kernel

\(\frac{1}{\sigma }\left( {1 + \frac{{\xi \left( {x - \mu } \right)}}{\sigma }} \right)^{\left( { - \frac{1}{\xi } - 1} \right)} \)

Support

\(x \ge \mu\) if \(\xi \ge 0\)

\(\mu \le x \le \mu - \sigma / \xi\) if \(\xi < 0\)

Mean

\(\mu  + \frac{\sigma }{{1 - \xi }}\) if \(\xi > 1\)

Variance

\(\frac{{\sigma ^2 }}{{(1 - \xi )^2 (1 - 2\xi )}}\) if \(\xi > 1/2\)

Main Uses

Calculation of tail probabilities

Density Function

%loggpdensity(x,xi,mu,sigma) is the log density at x given parameters \(\xi\), \(\mu\) and \(\sigma\)

CDF

%gpcdf(x,xi,mu,sigma) is the CDF of x given parameters \(\xi\), \(\mu\) and \(\sigma\)

%invgp(p,xi,mu,sigma\) is the inverse CDF of p given \(\xi\), \(\mu\) and \(\sigma\)