cdfpe3: Cumulative Distribution Function of the Pearson Type III...

In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

Cumulative Distribution Function of the Pearson Type III Distribution

Description

This function computes the cumulative probability or nonexceedance probability of the Pearson Type III distribution given parameters (\mu, \sigma, and \gamma) computed by parpe3. These parameters are equal to the product moments: mean, standard deviation, and skew (see pmoms). The cumulative distribution function is

F(x) = \frac{G\left(\alpha,\frac{Y}{\beta}\right)}{\Gamma(\alpha)} \mbox{,}

for \gamma \ne 0 and 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()), \Gamma is the complete gamma function, \xi is a location parameter, \beta is a scale parameter, \alpha 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

\alpha = 4/\gamma^2 \mbox{,}

\beta = \frac{1}{2}\sigma |\gamma| \mbox{,}

\xi = \mu - 2\sigma/\gamma \mbox{.}

Lastly, the function G(\alpha,x) is

G(\alpha,x) = \int_0^x t^{(a-1)} \exp(-t)\, \mathrm{d}t \mbox{.}

If \gamma = 0, the distribution is symmetrical and simply is the normal distribution with mean and standard deviation of \mu and \sigma, 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.

Usage

cdfpe3(x, para)

Arguments

x	A real value vector.
para	The parameters from parpe3 or vec2par.

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, v. 52, pp. 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

pdfpe3, quape3, lmompe3, parpe3

Examples

  lmr <- lmoms(c(123,34,4,654,37,78))
  cdfpe3(50,parpe3(lmr))

wasquith/lmomco documentation built on Nov. 13, 2024, 4:53 p.m.