cdfpe3 | R Documentation |
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.
cdfpe3(x, para)
x |
A real value vector. |
para |
The parameters from |
Nonexceedance probability (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, 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.
pdfpe3
, quape3
, lmompe3
, parpe3
lmr <- lmoms(c(123,34,4,654,37,78))
cdfpe3(50,parpe3(lmr))
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