| cdfgam {lmomco} | R Documentation |
This function computes the cumulative probability or nonexceedance probability
of the Gamma distribution given parameters (α and β) of the
distribution computed by pargam. The cumulative distribution
function of the distribution has no explicit form, but is expressed as an integral.
F(x) = frac{β^{-α}}{Γ(α)}int_0^x t^{α - 1} e^{-t/β} mbox{d}F mbox{,}
where F(x) is the nonexceedance probability for the quantile x. The parameters have the following interpretation in the R syntax; α is a shape parameter and β is a scale parameter.
cdfgam(x, para)
x |
A real value. |
para |
The parameters from pargam or similar. |
Nonexceedance probability (F) for x.
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, vol. 52, p. 105–124.
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.
lmr <- lmom.ub(c(123,34,4,654,37,78))
cdfgam(50,pargam(lmr))
# A manual demonstration of a gamma parent
G <- vec2par(c(0.6333,1.579),type='gam') # the parent
F1 <- 0.25 # nonexceedance probability
x <- quagam(F1,G) # the lower quartile (F=0.25)
a <- 0.6333 # gamma parameter
b <- 1.579 # gamma parameter
# compute the integral
xf <- function(t,A,B) { t^(A-1)*exp(-t/B) }
Q <- integrate(xf,0,x,A=a,B=b)
# finish the math
F2 <- Q$val*b^(-a)/gamma(a)
# check the result
if(abs(F1-F2) < 1e-8) print("yes")