Solved – a “log-F” distribution

Recently I have encountered the Wiener-Granger causality test. The statistic which is used is the logarithm of the ratio of two residual variances. It is well known that the ratio of two such (independent) Chi-squared random variables is F – distributed with parameters df1 (nominator) and df2 (denominator). What is the distribution of the log of a random variable which is F – distributed? Any reference would be highly appreciated.

It is a Type IV Generalized Logistic distribution.

Begin with the pdf of an $F(nu_1, nu_2)$ distribution ($nu_1$ is "df1" and $nu_2$ is "df2"), written $f(x)$. The pdf for the logarithm $y = log(x)$ will be the coefficient of $dy$ in

$$f(exp(y)), |mathrm d exp(y)| =frac{1}{Bleft(frac{nu _1}{2},frac{nu _2}{2}right)} left(frac{nu_1}{nu_2 + nu_1 exp(y)}right)^{nu_1/2} left(frac{nu_2}{nu_2 + nu_1 exp(y)}right)^{nu_2/2} mathrm d y.$$

Letting $mu = log(nu_1) – log(nu_2)$ exhibits the pdf in the form

$$frac{1}{Bleft(frac{nu _1}{2},frac{nu _2}{2}right)}frac{exp(-nu_2(y-mu))}{left(1 + exp(-(y-mu))right)^{(nu_1+nu_2)/2}} $$

which (upon setting $alpha = nu_1/2$ and $beta=nu_2/2$) is recognizable as being derived from

$$frac{exp(-beta y)}{left(1 + exp(-y)right)^{alpha+beta}}$$

via a shift of location to $mu = 2log(alpha) – 2log(beta)$ and normalization to unit probability.

Incidentally, if we let $l(y) = 1/(1 + exp(-y))$ be the logistic transformation (whose graph is a "sigmoid" that maps the real numbers to the interval $(0,1)$), this PDF can be presented in a more symmetric form

$$l(y)^alpha left(1 – l(y)right)^beta$$

which is reminiscent of the Beta distribution (and will be converted into it via $l$). Thus $alpha$ and $beta$ control the mean and variance of the distribution in a familiar way.

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