lmomln3 {lmomco}

L-moments of the 3-Parameter Log-Normal Distribution

Description

This function estimates the L-moments of the Log-Normal3 distribution given the parameters (\zeta, lower bounds; \mu_{\mathrm{log}}, location; and \sigma_{\mathrm{log}}, scale) from parln3. The distribution is the same as the Generalized Normal with algebraic manipulation of the parameters, and lmomco does not have truly separate algorithms for the Log-Normal3 but uses those of the Generalized Normal. The discussion begins with the later distribution.

The two L-moments in terms of the Generalized Normal distribution parameters (lmomgno) are

\lambda_1 = \xi + \frac{\alpha}{\kappa}[1-\mathrm{exp}(\kappa^2/2)] \mbox{, and}

\lambda_2 = \frac{\alpha}{\kappa}(\mathrm{exp}(\kappa^2/2)(1-2\Phi(-\kappa/\sqrt{2})) \mbox{,}

where \Phi is the cumulative distribution of the Standard Normal distribution. There are no simple expressions for \tau_3, \tau_4, and \tau_5, and numerical methods are used.

Let \zeta be 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 L-moments can be solved for using algorithms for the Generalized Normal.

Usage

lmomln3(para)

Arguments

para

The parameters of the distribution.

Value

An R list is returned.

lambdas

Vector of the L-moments. First element is \lambda_1, second element is \lambda_2, and so on.

ratios

Vector of the L-moment ratios. Second element is \tau, third element is \tau_3 and so on.

trim

Level of symmetrical trimming used in the computation, which is 0.

leftrim

Level of left-tail trimming used in the computation, which is NULL.

rightrim

Level of right-tail trimming used in the computation, which is NULL.

source

An attribute identifying the computational source of the L-moments: “lmomln3”.

Author(s)

W.H. Asquith

References

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.

See Also

parln3, cdfln3, pdfln3, qualn3, lmomgno

Examples

X <- exp(rnorm(10))
pargno(lmoms(X))$para
parln3(lmoms(X))$para