pdfpe3 {lmomco} | R Documentation |
This function computes the probability density
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 probability density function of the distribution for gamma ne 0 is
f(x) = frac{Y^{α -1} exp{-Y/β}} {β^α Γ(α)} mbox{,}
where f(x) is the probability density 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 probability density normal distribution with mean and standard deviation of μ and σ, respectively. Internally, the gamma = 0 condition is implemented by pnorm()
.
pdfpe3(x, para)
x |
A real value. |
para |
The parameters from parpe3 or similar. |
Probability density (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)) pe3 <- parpe3(lmr) x <- quape3(0.5,pe3) pdfpe3(x,pe3)