Fisher's z-distribution

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

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

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

where B is the beta function.

When the degrees of freedom becomes large ( $d_{1}, d_{2} \to \infty$ ), the distribution approaches normality with mean^[2]

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

and variance

σ_{x}^{2} = \frac{1}{2} (\frac{1}{d_{1}} + \frac{1}{d_{2}}) .

Related distribution