# Probability Density Function of the 4-Parameter Asymmetric Exponential Power Distribution 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 [REMOVE_ME]$$ f(x) = \frac{\kappa\,h}{\alpha(1+\kappa^2)\,\Gamma(1/h)}\,\mathrm{exp}\left( -\left[\kappa^{ \mathrm{sign}(x-\xi)}\left(\frac{|x-\xi|}{\alpha}\right)\,\right]^h \right) [REMOVE_ME_2]$$ 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 $-\infty < x < \infty$. If the $\tau_3$ of the distribution is zero (symmetrical), then the distribution is known as the Exponential Power (see `lmrdia46`). ## 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 $-\infty < x < \infty$. If the $\tau_3$ of the distribution is zero (symmetrical), then the distribution is known as the Exponential Power (see `lmrdia46`). ```r pdfaep4(x, para, paracheck=TRUE) ``` ## Arguments - `x`: A real value vector. - `para`: The parameters from `paraep4` or `vec2par`. - `paracheck`: A logical controlling whether the parameters and checked for validity. ## Returns Probability density ($f$) for $x$. ## Author(s) 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 ```r aep4 <- vec2par(c(1000,15000,0.5,0.4), type='aep4'); F <- nonexceeds(); x <- quaaep4(F,aep4); check.pdf(pdfaep4,aep4,plot=TRUE); ## Not run: 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); ## End(Not run) ```

pdfaep4 function