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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.

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  • $\begingroup$ For more on frequentist uses of the log-F see Jones MC. Families of distributions arising from distributions of order statistics. Test 2004; 13:1–43. The generalized log-F is also very useful in providing generalized conjugate prior distributions for logistic regression coefficients; see for example Greenland S. Prior data for non-normal priors. Stat Med 2007; 26:3578–90. $\endgroup$
    – user79852
    Jun 15, 2015 at 20:18

1 Answer 1

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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

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

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

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

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 $\mu = 2\log(\alpha) - 2\log(\beta)$ 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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  • $\begingroup$ Thank you very much for your immidiate and very useful response. I have another question. I am trying to find the statistically significant causal relationships between time series using the Wiener - Granger test. Therefore, I need the p - value for the test. Currently I am using R and I was wondering if you have any package (or function) in mind which is appropriate in this case. I found package "glogis" but its functions concern the type I genralized logistic and not type IV which you mentioned in your answer. Any thoughts? $\endgroup$
    – nikolaos
    Nov 26, 2014 at 21:22
  • $\begingroup$ Since the log of the statistic has an F distribution, then why not base your computation of the p-value on the log of the statistic? $\endgroup$
    – whuber
    Nov 26, 2014 at 21:28

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