quaaep4 | R Documentation |
This function computes the quantiles of the 4-parameter Asymmetric Exponential Power distribution given parameters (\xi
, \alpha
, \kappa
, and h
) of the distribution computed by paraep4
. The quantile function of the distribution given the cumulative distribution function F(x)
for F < F(\xi)
is
x(F) = \xi - \alpha\kappa\biggl[\gamma^{(-1)}\bigl((1+\kappa^2)F/\kappa^2,\; 1/h\bigr)\biggr]^{1/h}\mbox{,}
and for F \ge F(\xi)
is
x(F) = \xi + \frac{\alpha}{\kappa}\biggl[\gamma^{(-1)}\bigl((1+\kappa^2)(1-F),\; 1/h\bigr)\biggr]^{1/h} \mbox{,}
where x(F)
is the quantile x
for nonexceedance probability F
,
\xi
is a location parameter, \alpha
is a scale parameter,
\kappa
is a shape parameter, h
is another shape parameter, \gamma^{(-1)}(Z, shape)
is the inverse of the upper tail of the incomplete gamma function. The range of the distribution is -\infty < x < \infty
. The inverse upper tail of the incomplete gamma function is qgamma(Z, shape, lower.tail=FALSE)
in R. The mathematical definition of the upper tail of the incomplete gamma function shown in documentation for cdfaep4
. If the \tau_3
of the distribution is zero (symmetrical), then the distribution is known as the Exponential Power (see lmrdia46
).
quaaep4(f, para, paracheck=TRUE)
f |
Nonexceedance probability ( |
para |
The parameters from |
paracheck |
A logical controlling whether the parameters are checked for validity. Overriding of this check might be extremely important and needed for use of the quantile function in the context of TL-moments with nonzero trimming. |
Quantile value for nonexceedance probability F
.
W.H. Asquith
Asquith, W.H., 2014, Parameter estimation for the 4-parameter asymmetric exponential power distribution by the method of L-moments using R: Computational Statistics and Data Analysis, v. 71, pp. 955–970.
Delicado, P., and Goria, M.N., 2008, A small sample comparison of maximum likelihood, moments and L-moments methods for the asymmetric exponential power distribution: Computational Statistics and Data Analysis, v. 52, no. 3, pp. 1661–1673.
cdfaep4
, pdfaep4
, lmomaep4
, paraep4
para <- vec2par(c(0,1, 0.5, 2), type="aep4");
IQR <- quaaep4(0.75,para) - quaaep4(0.25,para);
cat("Interquartile Range=",IQR,"\n")
## Not run:
F <- c(0.00001, 0.0001, 0.001, seq(0.01, 0.99, by=0.01),
0.999, 0.9999, 0.99999);
delx <- 0.1;
x <- seq(-10,10, by=delx);
K <- .67
PAR <- list(para=c(0,1, K, 0.5), type="aep4");
plot(x,cdfaep4(x, PAR), type="n",
ylab="NONEXCEEDANCE PROBABILITY",
ylim=c(0,1), xlim=c(-20,20));
lines(x,cdfaep4(x,PAR), lwd=3);
lines(quaaep4(F, PAR), F, col=4);
PAR <- list(para=c(0,1, K, 1), type="aep4");
lines(x,cdfaep4(x, PAR), lty=2, lwd=3);
lines(quaaep4(F, PAR), F, col=4, lty=2);
PAR <- list(para=c(0,1, K, 2), type="aep4");
lines(x,cdfaep4(x, PAR), lty=3, lwd=3);
lines(quaaep4(F, PAR), F, col=4, lty=3);
PAR <- list(para=c(0,1, K, 4), type="aep4");
lines(x,cdfaep4(x, PAR), lty=4, lwd=3);
lines(quaaep4(F, PAR), F, col=4, lty=4);
## End(Not run)
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