cdfgam {lmomco}R Documentation

Cumulative Distribution Function of the Gamma Distribution

Description

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.

Usage

cdfgam(x, para)

Arguments

x A real value.
para The parameters from pargam or similar.

Value

Nonexceedance probability (F) for x.

Author(s)

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, 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.

See Also

quagam, pargam

Examples

  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")

[Package lmomco version 0.96.3 Index]