This function computes the cumulative probability or nonexceedance probability of the Generalized Pareto distribution given parameters (\(\xi\), \(\alpha\), and \(\kappa\)) computed by pargpa. The cumulative distribution function is
$$F(x) = 1 - \mathrm{exp}(-Y) \mbox{,}$$
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 nonexceedance probability 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.)
cdfgpa(x, para)Nonexceedance probability (\(F\)) for \(x\).
A real value vector.
The parameters from pargpa or vec2par.
W.H. Asquith
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.
Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press.
pdfgpa, quagpa, lmomgpa, pargpa
lmr <- lmoms(c(123, 34, 4, 654, 37, 78))
cdfgpa(50, pargpa(lmr))
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