# Cumulative Distribution Function of the 4-Parameter Asymmetric Exponential Power Distribution 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 [REMOVE_ME]$$ F(x) = \frac{\kappa^2}{(1+\kappa^2)} \; \gamma([(\xi - x)/(\alpha\kappa)]^h,\; 1/h)\mbox{,} [REMOVE_ME_2]$$ for $x < \xi$ and [REMOVE_ME]$$ F(x) = 1 - \frac{1}{(1+\kappa^2)} \; \gamma([\kappa(x - \xi)/\alpha]^h,\; 1/h)\mbox{,} [REMOVE_ME_2]$$ 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 and mathematically is [REMOVE_ME]$$ \gamma(Z, a) = \int_Z^\infty y^{a-1} \exp(-y)\, \mathrm{d}y \, /\, \Gamma(a)\mbox{.} [REMOVE_ME_2]$$ If the $\tau_3$ of the distribution is zero (symmetrical), then the distribution is known as the Exponential Power. ## 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 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. ```r 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. ## Returns Nonexceedance probability ($F$) 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 `pdfaep4`, `quaaep4`, `lmomaep4`, `paraep4` ## Examples ```r 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) ```

cdfaep4 function