This function computes the quantiles of the Log-Normal3 distribution given parameters (\(\zeta\), lower bounds; \(\mu_{\mathrm{log}}\), location; and \(\sigma_{\mathrm{log}}\), scale) of the distribution computed by parln3. The quantile function (same as Generalized Normal distribution, quagno) is
$$x = \Phi^{(-1)}(Y) \mbox{,} $$
where \(\Phi^{(-1)}\) is the quantile function of the Standard Normal distribution and \(Y\) is
$$
Y = \frac{\log(x - \zeta) - \mu_{\mathrm{log}}}{\sigma_{\mathrm{log}}}\mbox{,}
$$
where \(\zeta\) is the lower bounds (real space) for which \(\zeta < \lambda_1 - \lambda_2\) (checked in are.parln3.valid), \(\mu_{\mathrm{log}}\) be the mean in natural logarithmic space, and \(\sigma_{\mathrm{log}}\) be the standard deviation in natural logarithm space for which \(\sigma_{\mathrm{log}} > 0\) (checked in are.parln3.valid) is obvious because this parameter has an analogy to the second product moment. Letting \(\eta = \exp(\mu_{\mathrm{log}})\), the parameters of the Generalized Normal are \(\zeta + \eta\), \(\alpha = \eta\sigma_{\mathrm{log}}\), and \(\kappa = -\sigma_{\mathrm{log}}\). At this point, the algorithms (quagno) for the Generalized Normal provide the functional core.
qualn3(f, para, paracheck=TRUE)Quantile value for nonexceedance probability \(F\).
Nonexceedance probability (\(0 \le F \le 1\)).
The parameters from parln3 or vec2par.
A logical controlling whether the parameters are checked for validity. Overriding of this check might be extremely important and needed for use of the distribution quantile function in the context of TL-moments with nonzero trimming.
W.H. Asquith
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.
cdfln3, pdfln3, lmomln3, parln3, quagno
lmr <- lmoms(c(123,34,4,654,37,78))
qualn3(0.5,parln3(lmr))
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