pdfgpa: Probability Density Function of the Generalized Pareto Distribution

Description

This function computes the probability density of the Generalized Pareto distribution given parameters ($\xi$, $\alpha$, and $\kappa$) computed by pargpa. The probability density function is $$f(x) = \alpha^{-1} \exp(-(1-\kappa)Y) \mbox{,}$$ where $Y$ is $$Y = -\kappa^{-1} \log\left(1 - \frac{\kappa(x - \xi)}{\alpha}\right)\mbox{,}$$ for $\kappa \ne 0$, and $$Y = (x - \xi)/\alpha\mbox{,}$$ for $\kappa = 0$, where $f(x)$ is the probability density for quantile $x$, $\xi$ is a location parameter, $\alpha$ is a scale parameter, and $\kappa$ is a shape parameter. The range of $x$ is $\xi \le x \le \xi + \alpha/\kappa$ if $k > 0$; $\xi \le x < \infty$ if $\kappa \le 0$. Note that the shape parameter $\kappa$ parameterization of the distribution herein follows that in tradition by the greater L-moment community and others use a sign reversal on $\kappa$. (The evd package is one example.)

Usage

pdfgpa(x, para, paracheck=TRUE)

Value

Probability density ($f$) for $x$.

Arguments

x: A real value vector.
para: The parameters from pargpa or vec2par.
paracheck: A logical switch as to whether the validity of the parameters should be checked.

Author

W.H. Asquith

References

Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105--124, tools:::Rd_expr_doi("10.1111/j.2517-6161.1990.tb01775.x").

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.