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")