Fisher's z-distribution explained

Fisher's z-distribution is the statistical distribution of half the logarithm of an F-distribution variate:

z = (1)/(2) log F

It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto.^[1] Nowadays one usually uses the F-distribution instead.

The probability density function and cumulative distribution function can be found by using the F-distribution at the value of

x' = e ^{2x} . However, the mean and variance do not follow the same transformation.

The probability density function is^{[2] [3]}

f(x; d _{1,} d _{2)} = (2 d ^{d _{1/2}} _{1} d ^{d _{2/2}} _{2})/(B(d _{1/2,} d _{2/2)}) (e ^{d _{1} x})/(\left(d _{1} e ^{2} + d ^{(d _{1+d} _{2)/2}} _{2 \right)}) , where B is the beta function.

When the degrees of freedom becomes large (

d _{1,} d _{2} → inf ty ), the distribution approaches normality with mean^[2] \bar{x} = (1)/(2) \left( (1)/(d _{2}) - (1)/(d _{1}) \right) and variance\sigma ^{2} _{x} = (1)/(2) \left( (1)/(d _{1}) + (1)/(d _{2}) \right).

Related distribution

• If

X \sim \operatorname{FisherZ}(n,m) then e ^{2X} \sim \operatorname{F}(n,m) (F-distribution)

• If

X \sim \operatorname{F}(n,m) then \tfrac{ log X}{ 2 } \sim \operatorname{FisherZ}(n,m)

External links

• MathWorld entry

Notes and References

1. Fisher . R. A. . 1924 . On a Distribution Yielding the Error Functions of Several Well Known Statistics . Proceedings of the International Congress of Mathematics, Toronto . 2 . 805–813 . dead . https://web.archive.org/web/20110412083610/http://digital.library.adelaide.edu.au/coll/special//fisher/36.pdf . April 12, 2011 .
2. Leo A. Aroian . Leo A. Aroian . A study of R. A. Fisher's z distribution and the related F distribution . The Annals of Mathematical Statistics . 12 . 4 . December 1941 . 2235955 . 10.1214/aoms/1177731681 . 429–448. free .
3. Book: Charles Ernest Weatherburn . Charles Ernest Weatherburn . A first course in mathematical statistics. 1961 .