This function estimates the L-moments of the Polynomial Density-Quantile4 distribution given the parameters (\(\xi\), \(\alpha\), and \(\kappa\)) from parpdq4. The L-moments in terms of the parameters are
$$\lambda_1 = \xi\mbox{,}$$
$$\lambda_2 = \frac{\alpha}{\kappa} \bigl(1-\kappa^2\bigr)\, \mathrm{atanh}(\kappa)\mathrm{\ for\ } \kappa > 0\mbox{,}$$
$$\lambda_2 = \frac{\alpha}{\kappa} \bigl(1+\kappa^2\bigr)\, \mathrm{atan}(\kappa)\mathrm{\ for\ } \kappa < 0\mbox{,}$$
$$\tau_3 = 0 \mbox{, and}$$
$$\tau_4 = -\frac{1}{4} + \frac{5}{4\kappa}\biggl(\frac{1}{\kappa} - \frac{1}{\mathrm{atanh}(\kappa)} \biggr) \mathrm{\ for\ } \kappa > 0\mbox{,}$$
$$\tau_4 = -\frac{1}{4} - \frac{5}{4\kappa}\biggl(\frac{1}{\kappa} - \frac{1}{\mathrm{atan}(\kappa)} \biggr) \mathrm{\ for\ } \kappa < 0\mbox{,}$$
lmompdq4(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.
A numeric field connected to the ifailtext; a value of 0 indicates fully successful operation of the function.
A message, instead of a warning, about the internal operations or operational limits of the function.
An attribute identifying the computational source of the L-moments: “lmompdq4”.
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
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").
parpdq4, cdfpdq4, pdfpdq4, quapdq4