This function estimates the L-moments of the Polynomial Density-Quantile3 distribution given the parameters (\(\xi\), \(\alpha\), and \(\kappa\)) from parpdq3. The L-moments in terms of the parameters are
$$\lambda_1 = \xi + \alpha\bigl[(1+\kappa)\log(1+\kappa) - (1-\kappa)\log(1-\kappa) - \kappa\log(4)\bigr]\mbox{,}$$
$$\lambda_2 = \frac{\alpha(1-\kappa^2)}{(1-\kappa\tau_3)}\mbox{,}$$
$$\tau_3 = \frac{1}{\kappa} - \frac{1}{\mathrm{arctanh}(\kappa)} \mbox{, and}$$
$$\tau_4 = (5\tau_3/\kappa) - 1\mbox{.}$$
lmompdq3(para, paracheck=TRUE)An R
list is returned.
Vector of the L-moments. First element is \(\lambda_1\), second element is \(\lambda_2\), and so on.
Vector of the L-moment ratios. Second element is \(\tau\), third element is \(\tau_3\) and so on.
Level of symmetrical trimming used in the computation, which is 0.
Level of left-tail trimming used in the computation, which is NULL.
Level of right-tail trimming used in the computation, which is NULL.
An attribute identifying the computational source of the L-moments: “lmompdq3”.
The parameters of the distribution.
A logical switch as to whether the validity of the parameters should be checked. Default is paracheck=TRUE.
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.
Hosking, J.R.M., 2007, Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics: Journal of Statistical Planning and Inference, v. 137, no. 9, pp. 2870--2891, tools:::Rd_expr_doi("10.1016/j.jspi.2006.10.010").
Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press.
parpdq3, cdfpdq3, pdfpdq3, quapdq3