# Quantile Function Mixture Between the 4-Parameter Asymmetric Exponential Power and Kappa Distributions This function computes the quantiles of a mixture as needed between the 4-parameter Asymmetric Exponential Power (AEP4) and Kappa distributions given L-moments (`lmoms`). The quantile function of a two-distribution mixture is supported by `par2qua2` and is [REMOVE_ME]$$ x(F) = (1-w) \times A(F) + w \times K(F)\mbox{,} [REMOVE_ME_2]$$ where $x(F)$ is the mixture for nonexceedance probability $F$, $A(F)$ is the AEP4 quantile function (`quaaep4`), $K(F)$ is the Kappa quantile function (`quakap`), and $w$ is a weight factor. Now, the above mixture is only applied if the $\tau_4$ for the given $\tau_3$ is within the overlapping region of the AEP4 and Kappa distributions. For this condition, the $w$ is computed by proration between the upper Kappa distribution bound (same as the $\tau_3$ and $\tau_4$ of the Generalized Logistic distribution, see `lmrdia`) and the lower bounds of the AEP4. For $\tau_4$ above the Kappa, then the AEP4 is exclusive and conversely, for $\tau_4$ below the AEP4, then the Kappa is exclusive. The $w$ therefore is the proration [REMOVE_ME]$$ w = [\tau^{K}_4(\hat\tau_3) - \hat\tau_4] / [\tau^{K}_4(\hat\tau_3) - \tau^{A}_4(\hat\tau_3)]\mbox{,} [REMOVE_ME_2]$$ where $\hat\tau_4$ is the sample L-kurtosis, $\tau^{K}_4$ is the upper bounds of the Kappa and $\tau^{A}_4$ is the lower bounds of the AEP4 for the sample L-skew ($\hat\tau_3$). The parameter estimation for the AEP4 by `paraep4` can fall back to pure Kappa if argument `kapapproved=TRUE` is set. Such a fall back is unrelated to the mixture described here. ## Description This function computes the quantiles of a mixture as needed between the 4-parameter Asymmetric Exponential Power (AEP4) and Kappa distributions given L-moments (`lmoms`). The quantile function of a two-distribution mixture is supported by `par2qua2` and is $$ x(F) = (1-w) \times A(F) + w \times K(F)\mbox{,} $$ where $x(F)$ is the mixture for nonexceedance probability $F$, $A(F)$ is the AEP4 quantile function (`quaaep4`), $K(F)$ is the Kappa quantile function (`quakap`), and $w$ is a weight factor. Now, the above mixture is only applied if the $\tau_4$ for the given $\tau_3$ is within the overlapping region of the AEP4 and Kappa distributions. For this condition, the $w$ is computed by proration between the upper Kappa distribution bound (same as the $\tau_3$ and $\tau_4$ of the Generalized Logistic distribution, see `lmrdia`) and the lower bounds of the AEP4. For $\tau_4$ above the Kappa, then the AEP4 is exclusive and conversely, for $\tau_4$ below the AEP4, then the Kappa is exclusive. The $w$ therefore is the proration $$ w = [\tau^{K}_4(\hat\tau_3) - \hat\tau_4] / [\tau^{K}_4(\hat\tau_3) - \tau^{A}_4(\hat\tau_3)]\mbox{,} $$ where $\hat\tau_4$ is the sample L-kurtosis, $\tau^{K}_4$ is the upper bounds of the Kappa and $\tau^{A}_4$ is the lower bounds of the AEP4 for the sample L-skew ($\hat\tau_3$). The parameter estimation for the AEP4 by `paraep4` can fall back to pure Kappa if argument `kapapproved=TRUE` is set. Such a fall back is unrelated to the mixture described here. ```r quaaep4kapmix(f, lmom, checklmom=TRUE) ``` ## Arguments - `f`: Nonexceedance probability ($0 \le F \le 1$). - `lmom`: A L-moment object created by `lmoms` or similar. - `checklmom`: Should the `lmom` be checked for validity using the `are.lmom.valid` function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the $\tau_4$ and $\tau_3$ inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check. ## Returns Quantile value for nonexceedance probability $F$. ## 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. ## See Also `par2qua2`, `quaaep4`, `quakap`, `paraep4`, `parkap` ## Examples ```r ## Not run: FF <- c(0.0001, 0.0005, 0.001, seq(0.01,0.99, by=0.01), 0.999, 0.9995, 0.9999); Z <- qnorm(FF) t3s <- seq(0, 0.5, by=0.1); T4step <- 0.02 pdf("mixture_test.pdf") for(t3 in t3s) { T4low <- (5*t3^2 - 1)/4; T4kapup <- (5*t3^2 + 1)/6 t4s <- seq(T4low+T4step, T4kapup+2*T4step, by=T4step) for(t4 in t4s) { lmr <- vec2lmom(c(0,1,t3,t4)) # make L-moments for lmomco if(! are.lmom.valid(lmr)) next # for general protection kap <- parkap(lmr) if(kap$ifail == 5) next # avoid further work if numeric problems aep4 <- paraep4(lmr, method="A") X <- quaaep4kapmix(FF, lmr) if(is.null(X)) next # one last protection plot(Z, X, type="l", lwd=5, col=1, ylim=c(-15,15), xlab="STANDARD NORMAL VARIATE", ylab="VARIABLE VALUE") mtext(paste("L-skew =",lmr$ratios[3], " L-kurtosis = ",lmr$ratios[4])) # Now add two more quantile functions for reference and review # of the mixture. These of course would not be done in practice # only quaaep4kapmix() would suffice. if(! as.logical(aep4$ifail)) { lines(Z, qlmomco(F,aep4), lwd=2, col=2) } if(! as.logical(kap$ifail)) { lines(Z, qlmomco(F,kap), lwd=2, col=3) } message("t3=",t3," t4=",t4) # stout for a log file } } dev.off() ## End(Not run) ```

quaaep4kapmix function