Log-Logistic Distribution

The log-logistic distribution, also known as the Fisk distribution, is used to model growth where the growth rate starts high and decreases over time. It is commonly applied in survival analysis and reliability engineering.

The log-logistic distribution has a scale parameter $\alpha$ and a shape parameter $\beta$. The cumulative distribution function (CDF) is given by:

The probability density function (PDF) is:

The log-logistic distribution is defined for all positive real numbers (x > 0). Both its scale parameter $\alpha$ and shape parameter $\beta$ must be strictly positive to ensure a valid probability distribution.

The log-logistic distribution is used in various fields such as survival analysis, reliability engineering, and economics. It is particularly useful for modeling lifetimes and failure times where the hazard function is non-monotonic.

• In survival analysis, the log-logistic distribution is used to model patient survival times in medical studies, particularly in cases where the hazard rate first increases and then decreases over time.

• Reliability engineers employ this distribution to analyze and predict failure times of electronic components and mechanical systems throughout their lifecycle.

• In the field of hydrology, researchers utilize the log-logistic distribution to model flood frequencies and extreme precipitation events in watershed systems.

• Economists apply this distribution to describe income distributions and analyze patterns of wealth inequality across different populations.

• Environmental scientists use the log-logistic distribution to model the spatial and temporal spread of pollutants and to analyze contamination levels in various ecosystems.

Notable properties include:

• The hazard function is $h(x)=\frac{\beta x^{\beta -1}}{{\alpha }^{\beta }+x^{\beta }}$.

• The quantile function is $Q(p)=\alpha (p/(1-p){)}^{1/\beta }$.

• The distribution exhibits a unimodal hazard rate when $\beta >1$.

• If X follows a logistic distribution, then Y = exp(X) follows a log-logistic distribution.

• The log-logistic distribution is a special case of the Burr Type XII distribution.

• For large $\beta$, the distribution approaches a lognormal distribution.

The log-logistic distribution is implemented by the LogLogisticDistribution class. It has one constructor that takes two arguments. The first argument is the scale parameter, and the second argument is the shape parameter.

The following constructs the same log-logistic distribution with scale parameter 6.8 and shape parameter 4.1:

var logLogistic = new LogLogisticDistribution(6.8, 4.1);

Dim logLogistic = New LogLogisticDistribution(6.8, 4.1)

No code example is currently available or this language may not be supported.

let logLogistic = LogLogisticDistribution(6.8, 4.1)

The LogLogisticDistribution class has two specific properties, ScaleParameter and ShapeParameter, which return the scale and shape parameters of the distribution.

LogLogisticDistribution has one static (Shared in Visual Basic) method, Sample(Random), which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the scale and shape parameters of the distribution.

var random = new Pcg32();
double sample = LogLogisticDistribution.Sample(random, 6.8, 4.1);

Dim random = New Pcg32()
Dim sample = LogLogisticDistribution.Sample(random, 6.8, 4.1)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = LogLogisticDistribution.Sample(random, 6.8, 4.1)

The above example uses the MersenneTwister class to generate uniform random numbers.

For more information on the logistic distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 23. Wiley Series in Probability and Statistics.

• Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data. Wiley.

Other Resources