Fisk distribution (log-logistic)

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Have you ever wondered why certain situations seem to start off with a rapid increase and then after some time begin to slow down? The Fisk distribution, also called the log-logistic distribution, is a continuous probability distribution that models this phenomenon. The term ‘Fisk distribution’ is more common in economics; ‘log-logistic distribution’ is more common in other fields. This type of distribution has many applications and can be found in economics, hydrology, and biostatistics. Let’s take a closer look at what the Fisk Distribution is and how it can be used.

What is the Fisk Distribution?

fisk distribution pdf
The PDF of the Fisk distribution [1].

The Fisk distribution has a continuous probability density function (PDF) that describes data with an initial rapid growth followed by a gradual decrease. The distribution is named after P. Fisk, who proposed it to investigate weekly agriculture earnings in the 1950s [2].

The distribution It is basically a combination of two different distributions: the exponential distribution and gamma distribution. This means that it has two different parameters—one for the exponential part and one for the gamma part—which allows it to model complex phenomena. Additionally, it has some useful properties such as being unimodal, having positive skewness, and having heavy tails.

The distribution has two parameters, a scale parameter (λ) a shape parameter (κ). Both are positive numbers (alternate parameterizations include μ,s and α, β).

The expression  x ~ loglogistic(λ, κ) tells us that a random variable x follows a log-logistic (Fisk) distribution.

The probability density function (PDF) is given by:

fisk distribution PDF

The cumulative distribution function (CDF) is:

The survival function is:

fisk distribution survival function

The population mean and population variance is:

The median of x is 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 It 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.


[1] Image: Qwfp at en.wikipedia, CC BY-SA 3.0, via Wikimedia Commons

[2] Fisk, P. The Graduation of Income Distributions. Econometrica Vol. 29, No. 2 (Apr., 1961), pp. 171-185 (15 pages) Published By: The Econometric Society

[3] R Documentation. The Fisk Distribution. Retrieved April 5, 2023 from:

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