lmomco (version 1.7.3)

pdfaep4: Probability Density Function of the 4-p 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 $h$) of the distribution computed by paraep4. The probability density function of the distribution is

$$f(x) = \frac{\kappa\,h}{\alpha(1+\kappa^2)\,\Gamma(1/h)} \exp[ -(\kappa^{ \mathrm{sign}(x-\xi)}\,(\,|x-\xi|/\alpha)\,)^h ]$$

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 of the distribution is $-\infty < x < \infty$.

Usage

pdfaep4(x, para, paracheck=TRUE)

Arguments

x

A real value.

para

The parameters from paraep4 or similar.

paracheck

A logical controlling whether the parameters and checked for validity.

Value

• Probability density ($f$) for $x$.

References

Ayebo, A., and Kozubowski, T.J., 2003, An asymmetric generalization of Gaussian and Laplace laws: Journal of Probability and Statistical Science, v. 1, no. 2, pp. 187-210.

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, 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);

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);