< List of probability distributions > Fisher Z distribution
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]
Where
- ζ = location parameter,
- λ = scale parameter,
- α and γ = positive shape parameters α and γ,
- B( α, γ) = the beta function.
When ζ = 0 and e λ = 1, the distribution is called the standard Fisher Z distribution.
An alternate parameterization, given by [4] is
Prentice [5] parameterized the distribution as
Special cases include: Burr Type II distribution, Reversed Burr Type II distribution, logistic distribution, and hyperbolic secant distribution.
References
[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, Toronto. 2: 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.