lmomco (version 1.7.3)

cdfaep4: Cumulative Distribution Function of the 4-p Asymmetric Exponential Power Distribution

Description

This function computes the cumulative probability or nonexceedance probability of the 4-parameter Asymmetric Exponential Power distribution given parameters ($\xi$, $\alpha$, $\kappa$, and $h$) of the distribution computed by paraep4. The cumulative distribution function of the distribution is

For $x < \xi$,

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

and for $x \ge \xi$,

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

where $F(x)$ is the nonexceedance probability for quantile $x$, $\xi$ is a location parameter, $\alpha$ is a scale parameter, $\kappa$ is a shape parameter, $h$ is another shape parameter, $\gamma(Z, shape)$ is the upper tail of the incomplete gamma function. The upper tail of the incomplete gamma function is pgamma(Z, shape, lower.tail=FALSE) in Rand mathematically is

$$\gamma(Z, a) = \int_Z^\infty y^{a-1} \exp(-y) dy \; / \Gamma(a)\mbox{.}$$

Usage

cdfaep4(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

• Nonexceedance probability ($F$) 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

x <- -0.1
para <- vec2par(c(0, 100, 0.5, 4), type="aep4")
F <- cdfaep4(-.1,para)
estx <- quaaep4(F, para)
cat(c("F=",F,"and estx=",estx,""))

delx <- .1
x <- seq(-20,20, by=delx);
K <- 1;

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=range(x));
lines(x,cdfaep4(x,PAR), lwd=4);
lines(quaaep4(cdfaep4(x,PAR),PAR), cdfaep4(x,PAR), col=2)

PAR <- list(para=c(0,1, K, 1), type="aep4");
lines(x,cdfaep4(x, PAR), lty=2, lwd=4);
lines(quaaep4(cdfaep4(x,PAR),PAR), cdfaep4(x,PAR), col=2)

PAR <- list(para=c(0,1, K, 2), type="aep4");
lines(x,cdfaep4(x, PAR), lty=3, lwd=4);
lines(quaaep4(cdfaep4(x,PAR),PAR), cdfaep4(x,PAR), col=2)

PAR <- list(para=c(0,1, K, 4), type="aep4");
lines(x,cdfaep4(x, PAR), lty=4, lwd=4);
lines(quaaep4(cdfaep4(x,PAR),PAR), cdfaep4(x,PAR), col=2)