# quaaep4: Quantile Function of the 4-Parameter Asymmetric Exponential...

### Description

This function computes the quantiles of the 4-parameter Asymmetric Exponential Power distribution given parameters (ξ, α, κ, 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(ξ),

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

and for F ≥ F(ξ),

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

where x(F) is the quantile x for nonexceedance probability F, ξ is a location parameter, α is a scale parameter, κ is a shape parameter, h is another shape parameter, γ^{(-1)}(Z, shape) is the inverse of the upper tail of the incomplete gamma function. The range of the distribution is -∞ < x < ∞. 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

 1 quaaep4(f, para, paracheck=TRUE) 

### Arguments

 f Nonexceedance probability (0 ≤ F ≤ 1). para The parameters from paraep4 or similar. paracheck A logical controlling whether the parameters are checked for validity. Overriding of this check might be extremely important and needed for use of the quantile function in the context of TL-moments with nonzero trimming.

### Value

Quantile value for nonexceedance probability F.

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.

cdfaep4, pdfaep4, lmomaep4, paraep4

### Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 para <- vec2par(c(0,1, 0.5, 2), type="aep4"); IQR <- quaaep4(0.75,para) - quaaep4(0.25,para); cat("Interquartile Range=",IQR,"\n") ## Not run: 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); ## End(Not run) 

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