lmomco (version 2.4.14)

pdfaep4: Probability Density Function of the 4-Parameter Asymmetric Exponential Power Distribution

Description

This function computes the probability density of the 4-parameter Asymmetric Exponential Power distribution given parameters (\(\xi\), \(\alpha\), \(\kappa\), and \(h\)) computed by paraep4. The probability density function is $$f(x) = \frac{\kappa\,h}{\alpha(1+\kappa^2)\,\Gamma(1/h)}\, \mathrm{exp}\left( -\left[\kappa^{ \mathrm{sign}(x-\xi)}\left(\frac{|x-\xi|}{\alpha}\right)\,\right]^h \right) $$ where \(f(x)\) is the probability density for quantile \(x\), \(\xi\) is a location parameter, \(\alpha\) is a scale parameter, \(\kappa\) is a shape parameter, and \(h\) is another shape parameter. The range is \(-\infty < x < \infty\).

Usage

pdfaep4(x, para, paracheck=TRUE)

Value

Probability density (\(f\)) for \(x\).

Arguments

x

A real value vector.

para

The parameters from paraep4 or vec2par.

paracheck

A logical controlling whether the parameters and checked for validity.

Author

W.H. Asquith

References

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.

See Also

cdfaep4, quaaep4, lmomaep4, paraep4

Examples

aep4 <- vec2par(c(1000,15000,0.5,0.4), type='aep4');
F <- nonexceeds();
x <- quaaep4(F,aep4);
check.pdf(pdfaep4,aep4,plot=TRUE);
if (FALSE) {
delx <- .01;
x <- seq(-10,10, by=delx);
K <- 3;
PAR <- list(para=c(0,1, K, 0.5), type="aep4");
plot(x,pdfaep4(x, PAR), type="n",
     ylab="PROBABILITY DENSITY",
     ylim=c(0,0.6), xlim=range(x));
lines(x,pdfaep4(x,PAR), lwd=2);

PAR <- list(para=c(0,1, K, 1), type="aep4");
lines(x,pdfaep4(x, PAR), lty=2, lwd=2);

PAR <- list(para=c(0,1, K, 2), type="aep4");
lines(x,pdfaep4(x, PAR), lty=3, lwd=2);

PAR <- list(para=c(0,1, K, 4), type="aep4");
lines(x,pdfaep4(x, PAR), lty=4, lwd=2);
}