parpe3. These parameters are equal to the product moments: mean, standard deviation, and skew (see pmoms). The cumulative distribution function of the distribution for $\gamma > 0$ is$$F(x) = \frac{G\left(\alpha,\frac{x - \xi}{\beta}\right)}{\Gamma(\alpha)} \mbox{,}$$
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()), $\Gamma$ is the complete gamma function,
$\xi$ is a location parameter, $\beta$ is a scale parameter, and
$\alpha$ is a shape parameter. These three
$$\alpha = 4/\gamma^2 \mbox{,}$$ $$\beta = \frac{1}{2}\sigma |\gamma| \mbox{,}$$ $$\xi = \mu - 2\sigma/\gamma \mbox{.}$$
The function $G(\alpha,x)$ is $$G(\alpha,x) = \int_0^x t^{(a-1)} \mathrm{e}^{-t} \mathrm{d}t \mbox{.}$$
If $\gamma = 0$, the distribution is symmetrical and simply is the normal distribution with mean and standard deviation of $\mu$ 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.
cdfpe3(x, para)parpe3 or similar.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.
quape3, parpe3lmr <- lmom.ub(c(123,34,4,654,37,78))
cdfpe3(50,parpe3(lmr))Run the code above in your browser using DataLab