lmomco (version 2.4.14)

quaaep4kapmix: Quantile Function Mixture Between the 4-Parameter Asymmetric Exponential Power and Kappa Distributions

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.

Usage

quaaep4kapmix(f, lmom, checklmom=TRUE)

Value

Quantile value for nonexceedance probability \(F\).

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.

Author

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

if (FALSE) {
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()
}