Cumulative Distribution Function of the 4-Parameter Asymmetric Exponential Power Distribution

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Description

This function computes the cumulative probability or nonexceedance probability of the 4-parameter Asymmetric Exponential Power distribution given parameters (ξ, α, κ, and h) computed by paraep4. The cumulative distribution function is

F(x) = \frac{κ^2}{(1+κ^2)} \; γ([(ξ - x)/(ακ)]^h,\; 1/h)\mbox{,}

for x < ξ and

F(x) = 1 - \frac{1}{(1+κ^2)} \; γ([κ(x - ξ)/α]^h,\; 1/h)\mbox{,}

for x ≥ ξ, where F(x) is the nonexceedance probability for quantile x, ξ is a location parameter, α is a scale parameter, κ is a shape parameter, h is another shape parameter, and γ(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

γ(Z, a) = \int_Z^∞ y^{a-1} \exp(-y)\, \mathrm{d}y \, /\, Γ(a)\mbox{.}

Usage

1
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.

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

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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)

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