LogLogistic distribution | Vose Software

# LogLogistic distribution

Format: LogLogistic(a, b)

When Log(X) takes a Logistic distribution then X takes a LogLogistic distribution. Their parameters are related as follows:

EXP(Logistic(a,b)) = LogLogistic(1/b, EXP(a))

LogLogistic(a,1) is the standard LogLogistic distribution.

## Uses

The LogLogistic distribution has the same relationship to the Logistic distribution that the Lognormal distribution has to the Normal distribution. If you feel that a variable is driven by some process that is the product of a number of variables, then a natural distribution to use is the Lognormal because of Central Limit Theorem. However, if one or two of these factors could be dominant, or correlated, so that the distribution is less spread than a lognormal, then the LogLogistic may be an appropriate distribution to try.

The Loglogistic distribution, which has a heavier tail than the Gamma distribution, has been fit to quite a number of finance and insurance variables, for example: the duration of claim for income protection insurance (i.e. time until claimant returns to work); residuals for a time series regression of agricultural product values; insurance losses; natural catastrophe claims, etc.

From the explanation of the Logistic distribution you can see that the limiting distribution of the geometric mean of the minimum and maximum samples from an exponential family distribution will take a LogLogistic distribution. There are relatively few applications of the LogLogistic distribution that directly use its underpinning mathematical model.

The parameter a defines the shape of the distribution, and the distribution's spread is proportional to b. Descriptions and applications of the LogLogistic model may be found in Bacon (1993), Diekmann (1992), Little, Adams, and Anderson (1994), Nandram (1989), and Singh, Lee, and George (1988).

The LogLogistic is also sometimes known as the Fisk distribution.

## ModelRisk functions added to Microsoft Excel for the LogLogistic distribution

VoseLogLogistic generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseLogLogisticObject constructs a distribution object for this distribution.

VoseLogLogisticProb returns the probability density or cumulative distribution function for this distribution.

VoseLogLogisticProb10 returns the log10 of the probability density or cumulative distribution function.

VoseLogLogisticFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseLogLogisticFitObject constructs a distribution object of this distribution fitted to data.

VoseLogLogisticFitP returns the parameters of this distribution fitted to data.