Fisher's z-distribution

Short description: Statistical distribution

Fisher's z

Probability density function
Parameters	[math]\displaystyle{ d_1\gt 0,\ d_2\gt 0 }[/math] deg. of freedom
Support	[math]\displaystyle{ x \in (-\infty; +\infty)\! }[/math]
PDF	[math]\displaystyle{ \frac{2d_1^{d_1/2}d_2^{d_2/2}}{B(d_1/2,d_2/2)}\frac{e^{d_1x}}{\left(d_1e^{2x}+d_2\right)^{\left(d_1+d_2\right)/2}}\! }[/math]
Mode	[math]\displaystyle{ 0 }[/math]

Ronald Fisher

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

[math]\displaystyle{ z = \frac 1 2 \log F }[/math]

It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto. 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 [math]\displaystyle{ x' = e^{2x} \, }[/math]. However, the mean and variance do not follow the same transformation.

The probability density function is

[math]\displaystyle{ f(x; d_1, d_2) = \frac{2d_1^{d_1/2} d_2^{d_2/2}}{B(d_1/2, d_2/2)} \frac{e^{d_1 x}}{\left(d_1 e^{2 x} + d_2\right)^{(d_1+d_2)/2}}, }[/math]

where B is the beta function.

When the degrees of freedom becomes large ([math]\displaystyle{ d_1, d_2 \rightarrow \infty }[/math]), the distribution approaches normality with mean

[math]\displaystyle{ \bar{x} = \frac 1 2 \left( \frac 1 {d_2} - \frac 1 {d_1} \right) }[/math]

and variance

[math]\displaystyle{ \sigma^2_x = \frac 1 2 \left( \frac 1 {d_1} + \frac 1 {d_2} \right). }[/math]

Related distribution

• If [math]\displaystyle{ X \sim \operatorname{FisherZ}(n,m) }[/math] then [math]\displaystyle{ e^{2X} \sim \operatorname{F}(n,m) \, }[/math] (F-distribution)
• If [math]\displaystyle{ X \sim \operatorname{F}(n,m) }[/math] then [math]\displaystyle{ \tfrac{\log X}{2} \sim \operatorname{FisherZ}(n,m) }[/math]

External links

• MathWorld entry