| cdfpe3 {lmomco} | R Documentation |
This function computes the cumulative probability or nonexceedance probability
of the Pearson Type III distribution given parameters (μ, σ,
and gamma) of the distribution computed
by parpe3. These parameters are equal to the product moments: mean, standard deviation, and skew (see pmoms). The cumulative distribution function of the distribution for gamma ne 0 is
F(x) = frac{G(α,frac{Y}{β})}{Γ(α)} mbox{,}
where F(x) is the nonexceedance probability for quantile x,
G is defined below and is related to the incomplete gamma function of R (pgamma()), Γ is the complete gamma function,
xi is a location parameter, β is a scale parameter,
α is a shape parameter, and Y = x - xi if gamma > 0 and Y = xi - x if gamma < 0. These three “new” parameters are related to the product moments by
α = 4/gamma^2 mbox{,}
β = frac{1}{2}σ |gamma| mbox{,}
xi = μ - 2σ/gamma mbox{.}
The function G(α,x) is
G(α,x) = int_0^x t^{(a-1)} mathrm{e}^{-t} mathrm{d}t mbox{.}
If gamma = 0, the distribution is symmetrical and simply is the normal distribution with mean and standard deviation of μ and σ, respectively. Internally, the gamma = 0 condition is implemented by pnorm(). If gamma > 0, the distribution is right-tail heavy, and F(x) is the returned nonexceedance probability. On the other hand if gamma < 0, the distribution is left-tail heavy and 1-F(x) is the actual nonexceedance probability that is returned.
cdfpe3(x, para)
x |
A real value. |
para |
The parameters from parpe3 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)) cdfpe3(50,parpe3(lmr))