The Fisk distribution, also called the log-logistic distribution, is a continuous probability distribution with 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 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 Fisk Distribution’s Probability Density Function
The probability distribution 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.
Al-Shomrani, Shawky, Arif & Aslam. Log-logistic distribution for survival data analysis using MCMC. SpringerPlus (2016) 5:1774. DOI 10.1186/s40064-016-3476-7. Retrieved from https://link.springer.com/content/pdf/10.1186/s40064-016-3476-7.pdf on May 18, 2018
Leemis, Larry. Log Logistic Distribution. Retrieved form
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Loglogistic.pdf on May 18, 2018
World Heritage Encyclopedia, Fisk Distribution. Retrieved from http://self.gutenberg.org/articles/fisk_distribution on May 6, 2018