parkap: Estimate the Parameters of the Kappa Distribution

Description

This function estimates the parameters of the Kappa 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 lmomkap, but of relevance to this documentation, the upper bounds of L-kurtosis ($\tau_4$) and a function of L-skew ($\tau_3$) is given by $$\tau_4 < \frac{5\tau_3^2+1}{6}$$ This bounds is equal to the Generalized Logistic distribution (parglo) and failure occurs if this upper bounds is exceeded. However, the argument snap.tau4, if set, will set $\tau_4$ equal to the upper bounds of $\tau_4$ of the distribution to the relation above. This value of $\tau_4$ should be close enough numerically The argument nudge.tau4 is provided to offset $\tau_4$ downward just a little. This keeps the relation operator as “$<$” in the bounds above to match Hosking's tradition as his sources declare “$\ge$” as above the GLO. The nudge here hence is not zero, which is a little different compared to the conceptually similar snapping in paraep4.

Usage

parkap(lmom, checklmom=TRUE,
             snap.tau4=FALSE, nudge.tau4=sqrt(.Machine$double.eps), ...)

Value

An R

list is returned.

type: The type of distribution: kap.
para: The parameters of the distribution.
source: The source of the parameters: “parkap”.
support: The support (or range) of the fitted distribution.
ifail: A numeric failure code.
ifailtext: A text message for the failure code.

Arguments

lmom: An L-moment object created by lmoms or vec2lmom.
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.
snap.tau4: A logical to “snap” the $\tau_4$ downwards to the lower boundary if the given $\tau_4$ is greater than the boundary described as above.
nudge.tau4: An offset to the snapping of $\tau_4$ intended to move $\tau_4$ just below the upper bounds. (The absolute value of the nudge is made internally to ensure only downward adjustment by a subtraction operation.)
...: Other arguments to pass.

Author

W.H. Asquith

References

Hosking, J.R.M., 1994, The four-parameter kappa distribution: IBM Journal of Reserach and Development, v. 38, no. 3, pp. 251--258.