Shifted Log-logistic Distribution - Definition

Definition

The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter . Thus if has a log-logistic distribution then has a shifted log-logistic distribution. So has a shifted log-logistic distribution if has a logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the (unshifted) log-logistic.

The properties of this distribution are straightforward to derive from those of the log-logistic distribution. However, an alternative parameterisation, similar to that used for the generalized Pareto distribution and the generalized extreme value distribution, gives more interpretable parameters and also aids their estimation.

In this parameterisation, the cumulative distribution function of the shifted log-logistic distribution is

for, where is the location parameter, the scale parameter and the shape parameter. Note that some references use to parameterise the shape.

The probability density function is

f(x; \mu,\sigma,\xi) = \frac{\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-(1/\xi +1)}} {\sigma\left^2},

again, for

The shape parameter is often restricted to lie in, when the probability density function is bounded. When, it has an asymptote at . Reversing the sign of reflects the pdf and the cdf about .

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