lmomln3: 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)

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”.

Arguments

para: The parameters of the distribution.

Author

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.

Examples

Run this code

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

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