A.20 Fisher’s z distribution
X has a z distribution on and degrees of freedom, denoted
if , or equivalently if it has the density
Another definition is that if , then
The mean and variance are easily deduced from those of the log chi-squared distribution; they are
where is (as above) the digamma function, or approximately
The mode is zero.
Unless and are very small, the distribution of z is approximately normal. The z distribution was introduced by Fisher (1924).