Probability Distributions > Fisk distribution
The Fisk distribution, also called the log-logistic distribution, is a continuous probability distribution that models the phenomenon of certain situations that start off with a rapid increase and then after some time begin to slow down. The distribution is named after P. Fisk, who proposed it to investigate weekly agriculture earnings in the 1950s [1].
The Fisk distribution has many applications: from modeling the distribution of wealth or income in economics, to stream flow rates in hydrology to the lifetime of an organism in biostatistics. It is especially useful in modeling situations where the rate something is happening increases initially, and then after some time begins to decrease. The term ‘Fisk distribution’ is primarily used in economics; in other fields, ‘log-logistic distribution’ is more common.
The Fisk Distribution Probability Density Function
The Fisk distribution has two parameters, a scale parameter and a shape parameter. Both are positive numbers. To say that a random variable x has the log-logistic distribution, we write x ~ loglogistic(λ κ). Here λ is the scale parameter and κ is the shape parameter; both, again, are positive. The probability density function (pdf) is given by: 
where λ is the scale parameter and κ is the shape parameter.
Other Important Functions
The cumulative distribution function, on the support of X, is given by: 
The survival function, on the support of X, is given by: 
The population mean and variance, for a data set that follows the fisk distribution, is given by: 
The median of x is just 1/λ. As its name suggests, the log-logistic distribution is closely related to the logistic distribution. If a random variable x is distributed log-logistically with parameters λ and κ, then log(x) follows a logistic distribution parameters λ and κ. It is a special case of the four-parameter generalized beta II distribution [3].
Where is the Fisk distribution used?
The Fisk distribution is useful in many different fields. In economics, it can be used to model the wealth or income of people within a society since these tend to follow this type of pattern (i.e., initially increasing before tapering off). In hydrology, it can be used to model stream flow rates since they too often follow this type of pattern (i.e., initially increasing before decreasing). Finally, in biostatistics, it can be used to model the lifetimes of organisms since organisms tend to have an initial period of rapid growth followed by a period of gradual decline as they age.
References
[1] Fisk, P. The Graduation of Income Distributions. Econometrica Vol. 29, No. 2 (Apr., 1961), pp. 171-185 (15 pages) Published By: The Econometric Society. [2] Image: Qwfp at en.wikipedia, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons [3] R Documentation. The Fisk Distribution. Retrieved April 5, 2023 from: https://search.r-project.org/CRAN/refmans/VGAM/html/fiskUC.html