Fisher z-distribution

A continuous probability distribution on the real line with density

$$ f ( x) = $$

$$ = \ 2m _ {1} ^ {m _ {1} /2 } m _ {2} ^ {m _ {2} /2 } \frac{\Gamma ( ( m _ {1} + m _ {2} )/2) e ^ {m _ {1} x } }{\Gamma ( {m _ {1} /2 } ) \Gamma ( {m _ {2} /2 } ) ( m _ {1} e ^ {2x} + m _ {2} ) ^ {( m _ {1} + m _ {2} )/2 }} . $$

The parameters $ m _ {1} , m _ {2} \geq 1 $ are called the degrees of freedom. The characteristic function has the form

$$ \phi ( t) = \ \left ( \frac{m _ {2} }{m _ {1} } \right ) ^ { {{it } /2 } } \frac{\Gamma ( {( m _ {1} + it)/2 } ) \Gamma ( {( m _ {2} - it)/2 } ) }{\Gamma ( { {m _ {1} } /2 } ) \Gamma ( { {m _ {2} } /2 } ) } . $$

The mathematical expectation and the variance are equal to $ ( 1/m _ {1} - 1/m _ {2} )/2 $ and $ ( 1/m _ {1} + 1/m _ {2} )/2 $, respectively.

If the random variable $ F $ has the Fisher $ F $-distribution with $ m _ {1} $ and $ m _ {2} $ degrees of freedom, then the quantity $ z = ( \mathop{\rm log} F)/2 $ has the Fisher $ z $-distribution with $ m _ {1} $ and $ m _ {2} $ degrees of freedom. Along with the Fisher $ F $-distribution, known as the distribution of the dispersion proportion, the Fisher $ z $-distribution was originally introduced in the analysis of variance by R.A. Fisher (1924). His intention was that the $ z $-distribution should be the basic distribution for testing statistical hypotheses in the analysis of variance. The Fisher $ z $-distribution was tabulated at the same time, and the first research was concerned with the statistic $ z $, although in modern mathematical statistics one uses the simpler statistic $ F $.

References

[1]	R.A. Fisher, "On a distribution yielding the error functions of several well-known statistics" , Proc. Internat. Congress mathematicians (Toronto 1924) , 2 , Univ. Toronto Press (1928) pp. 805–813

Comments

The dispersion proportion is also called the variance ratio.