lmomco (version 1.7.3)

quaaep4: Quantile Function of the 4-p Asymmetric Exponential Power Distribution

Description

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 $F(x)$ is

For $F < F(\xi)$,

$$x(F) = \xi - \alpha\kappa[\gamma^{(-1)}((1+\kappa^2)F/\kappa^2,\; 1/h)]^{1/h}$$

and for $F \ge F(\xi)$,

$$x(F) = \xi + \frac{\alpha}{\kappa}[\gamma^{(-1)}((1+\kappa^2)(1-F),\; 1/h)]^{1/h}$$

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.

Usage

quaaep4(f, para, paracheck=TRUE)

Arguments

f

Nonexceedance probability ($0 \le F \le 1$).

para

The parameters from paraep4 or similar.

paracheck

A logical controlling whether the parameters and checked for validity.

Value

• Quantile value for nonexceedance probability $F$.

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

pdfaep4, cdfaep4, paraep4

Examples

para <- vec2par(c(0,1, 0.5, 2), type="aep4");
IQR <- quaaep4(0.75,para) - quaaep4(0.25,para);
cat("Interquartile Range=",IQR,"")

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);