# cdfaep4: Cumulative Distribution Function of the 4-Parameter... In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

 cdfaep4 R Documentation

## Cumulative Distribution Function of the 4-Parameter 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) computed by paraep4. The cumulative distribution function is

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

for x < \xi and

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

for x \ge \xi, 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, and \gamma(Z, s) is the upper tail of the incomplete gamma function for the two arguments. The upper tail of the incomplete gamma function is pgamma(Z, shape, lower.tail=FALSE) in R and mathematically is

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

If the \tau_3 of the distribution is zero (symmetrical), then the distribution is known as the Exponential Power.

### Usage

cdfaep4(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.

### Value

Nonexceedance probability (F) for x.

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.

pdfaep4, quaaep4, lmomaep4, paraep4

### Examples

x <- -0.1
para <- vec2par(c(0, 100, 0.5, 4), type="aep4")
FF <- cdfaep4(-.1,para)
cat(c("F=",FF,"  and estx=",quaaep4(FF, para),"\n"))
## Not run:
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",ylim=c(0,1), xlim=range(x),
ylab="NONEXCEEDANCE PROBABILITY");
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)
## End(Not run)


lmomco documentation built on May 29, 2024, 10:06 a.m.