SPEI

Obtaining location (\(\mu\)), scale (\(\sigma\)), and shape (\(\kappa\)) from L-moments (see J. R. M. Hosking and James R. Wallis (): Regional Frequency Analysis: An Approach Based on L-Moments):

\[k = -\tau_3\]

\[\sigma = \frac{\lambda_2 \sin(\kappa \pi)}{\kappa \pi}\]

\[\mu = \lambda_1 - \frac{\sigma}{\kappa} \left[ 1 - \frac{\kappa \pi}{\sin(\kappa \pi)} \right]\]

The shifted log-logistic distribution has a three parameters cumulative distribution function defined as:

\[F(x; \mu, \sigma, \kappa) = \frac{1}{1 + \left[ 1 + \kappa \frac{x - \mu}{\sigma} \right]^{-\frac{1}{\kappa}} }\]

The logistic distribution has a three parameters cumulative distribution function defined as:

\[F(x; \mu, \sigma) = \frac{1}{1 + e^{-\frac{x - \mu}{\sigma}}}\]