Fisher Z distribution (Beta-logistic)

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The Fisher Z distribution (also called the beta-logistic distribution) is a four-parameter univariate and unimodal continuous distribution with infinite support. It can provide a better fit than the normal distribution for certain types of data, such as heat destruction of bacteria in food protection studies [1].

The Fisher Z distribution was first introduced by R. A. Fisher in 1924 [2]. It has been “rediscovered” since that date by many authors and goes by a variety of other names, including:

  • Beta-prime exponential distribution,
  • Exponential generalized beta prime distribution,
  • Exponential generalized beta type II distribution,
  • Generalized F distribution,
  • Generalized Gompertz-Verhulst type II distribution,
  • Generalized logistic type IV distribution,
  • Log-F distribution,
  • Prentice distribution.

Fisher Z distribution properties

Although the Fisher Z distribution has many parameterizations of its probability density function (PDF), one of the most straightforward is [3]

fisher z distribution PDF


When ζ = 0 and e λ = 1, the distribution is called the standard Fisher Z distribution.

An alternate parameterization given by [4] is

alternate fisher z pdf

Prentice [5] also offers another parameterization of the distribution.

Special cases of the Fisher Z distribution include: Burr Type II distribution, Reversed Burr Type II distribution, logistic distribution, and hyperbolic secant distribution.


[1] Kilsby, et al. Bacterial thermal death kinetics based on probability distributions: the heat destruction of Clostridium botulinum and Salmonella Bedford. J Food Prot 2000 Sep;63(9):1197-203. doi: 10.4315/0362-028x-63.9.1197

[2]  Fisher, R. A. (1924). “On a Distribution Yielding the Error Functions of Several Well Known Statistics” (PDF). Proceedings of the International Congress of Mathematics, Toronto2: 805–813.

[3] Crooks, G. (2019). Field Guide to Continuous Probability Distributions.

[4] Leo A. Aroian (December 1941). “A study of R. A. Fisher’s z distribution and the related F distribution”. The Annals of Mathematical Statistics. 12 (4): 429–448. doi:10.1214/aoms/1177731681. JSTOR 2235955.

[5] Prentice, R. L. (1975) Discrimination among some parametric models. Biometrika, 62(3):607-614.

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