This function estimates the parameters of the 3-parameter Student t distribution given the L-moments of the data in an L-moment object such as that returned by lmoms. The relations between distribution parameters and L-moments are seen under lmomst3. The largest value of \(\nu\) recognized is \(10^5.5\), which is the near the Normal distribution and the smallest value recognized is \(1.001\), which is near the Cauchy. As \(\nu \rightarrow 1\) the distribution limits to the Cauchy, but the implementation here does not switch over to the Cauchy. Therefore in lmomco \(1.001 \le \nu \le 10^5.5\). The \(\nu\) is the “degrees of freedom” parameter that is well-known with the 1-parameter Student t distribution. The \(nu\) limits are studied in the inst/doc/t4t6/studyST3.R script and the theoTLmoms function and its performance on the quantile function of the distribution provide the guidance including range of numerically computable \(\tau_6\). The \(\tau_4\) value can be set as low as \(0.1226\) as short-hand for the true lower L-kurtosis limit, which is that of the Normal (\(30/\pi \times \mathrm{atan}(\sqrt{2}) - 9 = 0.1226017\) and additional decimals). Internally, the given \(0.1226 \le \tau_4 \le 0.1226017\) is snapped to that of the Normal with an internal small positive nudge up. The \(\tau_4 > 0.998\) are set to \(\tau_4 = 0.998\).
parst3(lmom, checklmom=TRUE, ...)An R
list is returned.
The type of distribution: st3.
The parameters of the distribution.
The returned list of the uniroot() call to estimate \(\nu\).
The source of the parameters: “parst3”.
An L-moment object created by lmoms or vec2lmom.
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.
Other arguments to pass.
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.
lmomst3, cdfst3, pdfst3, quast3
parst3(vec2lmom(c(10, 2, 0, 0.1226)))$para
parst3(vec2lmom(c(10, 2, 0, 0.14 )))$para
parst3(vec2lmom(c(10, 2, 0, 0.4 )))$para
parst3(vec2lmom(c(10, 2, 0, 0.9 )))$para
parst3(vec2lmom(c(10, 2, 0, 0.998 )))$para
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