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 \(k<0\).
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.