This function computes the cumulative probability or nonexceedance probability of the Gamma distribution given parameters (\(\alpha\) and \(\beta\)) computed by pargam. The cumulative distribution function has no explicit form but is expressed as an integral:
$$F(x) = \frac{\beta^{-\alpha}}{\Gamma(\alpha)}\int_0^x t^{\alpha - 1}
\exp(-t/\beta)\; \mbox{d}t \mbox{,}$$
where \(F(x)\) is the nonexceedance probability for the quantile \(x\), \(\alpha\) is a shape parameter, and \(\beta\) is a scale parameter.
Alternatively, a three-parameter version is available following the parameterization of the Generalized Gamma distribution used in the gamlss.dist package and is
$$F(x) =\frac{\theta^\theta\, |\nu|}{\Gamma(\theta)}\int_0^x \frac{z^\theta}{x}\,\mathrm{exp}(-z\theta)\; \mbox{d}x \mbox{,}$$
where \(z =(x/\mu)^\nu\), \(\theta = 1/(\sigma^2\,|\nu|^2)\) for \(x > 0\), location parameter \(\mu > 0\), scale parameter \(\sigma > 0\), and shape parameter \(-\infty < \nu < \infty\). The three parameter version is automatically triggered if the length of the para element is three and not two.
cdfgam(x, para)Nonexceedance probability (\(F\)) for \(x\).
A real value vector.
The parameters from pargam 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.
Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press.
pdfgam, quagam, lmomgam, pargam