Fisher's z-distribution

Fisher's z

Parameters	$d_{1}>0,\ d_{2}>0$ deg. of freedom
Support	$x\in (-\infty ;+\infty )\!$
PDF	${\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}z}}{\left(d_{1}e^{2z}+d_{2}\right)^{\left(d_{1}+d_{2}\right)/2}}}\!$
Mode	$0$

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

$z={\frac {1}{2}}\log F$

It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled "On a distribution yielding the error functions of several well-known statistics" (Proceedings of the International Congress of Mathematics, Toronto, 2: 805-813 (1924). Nowadays one usually uses the F distribution instead.

The probability density function and cumulative distribution function can be found be 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

$f(x;d_{1},d_{2})={\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{(d_{1}+d_{2})/2}}},$

where B is the beta function.

When the degrees of freedom becomes large ($n_{1},n_{2}\rightarrow \infty$) the distribution approach normality with mean

${\bar {x}}=(1/d_{2}-1/d_{1})/2$

and variance

$\sigma _{x}^{2}=(1/d_{1}+1/d_{2})/2.$

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

References

• 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 pdf copy

External links

• MathWorld entry