pdfpe3 {lmomco}R Documentation

Probability Density Function of the Pearson Type III Distribution

Description

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().

Usage

pdfpe3(x, para)

Arguments

x A real value.
para The parameters from parpe3 or similar.

Value

Probability density (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

cdfpe3, quape3, parpe3

Examples

  lmr <- lmom.ub(c(123,34,4,654,37,78))
  pe3 <- parpe3(lmr)
  x <- quape3(0.5,pe3)
  pdfpe3(x,pe3)

[Package lmomco version 0.96.3 Index]