See http://en.wikipedia.org/wiki/Log-normal_distribution for an introduction to the lognormal distribution.Definition
Parameters (3): $\xi$ (location), $\alpha$ (scale), $k$ (shape).
Range of $x$: $-\infty < x \le \xi + \alpha / k$ if $k>0$;
$-\infty < x < \infty$ if $k=0$;
$\xi + \alpha / k \le x < \infty$ if $k<0$.< p="">
Probability density function:
$$f(x) = \frac{e^{ky-y^2/2}}{\alpha \sqrt{2\pi}}$$
where $y = -k^{-1}\log{1 - k(x - \xi)/\alpha}$ if $k \ne 0$,
$y = (x-\xi)/\alpha$ if $k=0$.
Cumulative distribution function:
$$F(x) = \Phi(x)$$
where
$\Phi(x)=\int_{-\infty}^x \phi(t)dt$.
Quantile function:
$x(F)$ has no explicit analytical form.
$k=0$ is the Normal distribution with parameters $\xi$ and $alpha$.
L-moments
L-moments are defined for all values of $k$.
$$\lambda_1 = \xi + \alpha(1 - e^{k^2/2})/k$$
$$\lambda_2 = \alpha/k e^{k^2/2} [1 - 2 \Phi(-k/\sqrt{2})]$$
There are no simple expressions for the L-moment ratios $\tau_r$ with $r \ge 3$.
Here we use the rational-function approximation given in Hosking and Wallis (1997, p. 199).
Parameters
The shape parameter $k$ is a function of $\tau_3$ alone.
No explicit solution is possible.
Here we use the approximation given in Hosking and Wallis (1997, p. 199).
Given $k$, the other parameters are given by
$$\alpha = \frac{\lambda_2 k e^{-k^2/2}}{1-2 \Phi(-k/\sqrt{2})}$$
$$\xi = \lambda_1 - \frac{\alpha}{k} (1 - e^{k^2/2})$$
Lmom.lognorm and par.lognorm accept input as vectors of equal length. In f.lognorm, F.lognorm, invF.lognorm and rand.lognorm parameters (xi, alfa, k) must be atomic.