cdfpe3: 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)

Value

Nonexceedance probability ($F$) for $x$.

Arguments

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

Author

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.

Examples

Run this code

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

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