Generalizing the Pareto to the log-Pareto model and statistical inference (Q626281)

The generalized log-Pareto (GLPD) CDF is defined as \[ L_{\gamma,\beta,\sigma}(x)=1-( 1+\gamma\beta^{-1}\log(x\sigma^{-1}))^{-1/\gamma} \] for \(x>\sigma\) if \(\gamma\geq 0\) and for \(\sigma<x<\sigma e^{\beta/| \gamma| }\) if \(\gamma<0\). Here \(\beta>0\) and \(\gamma\) are the shape parameters, and \(\sigma>0\) is the scale parameter. It is shown that all continuous peak-over-threshold stable distributions are of GLPD form. Maximum likelihood estimates and quick estimates based on empirical quantiles are considered for the parameters of the GLPD. Results of simulations and applications to biologic and internet data are presented.