The inverse chi--squared distribution
with \(df = \nu \geq 0\) degrees of
freedom implemented here has density
$$f(x; \nu) = \frac{ 2^{-\nu / 2} x^{-\nu/2 - 1}
e^{-1 / (2x)} }{ \Gamma(\nu / 2) }, $$
where \(x > 0\), and
\(\Gamma\) is the gamma function.
The mean of \(Y\) is \(1 / (\nu - 2)\) (returned as the fitted
values), provided \(\nu > 2\).
That is, while the expected information matrices used here are
valid in all regions of the parameter space, the regularity conditions
for maximum likelihood estimation are satisfied only if \(\nu > 2\).
To enforce this condition, choose
link = logoff(offset = -2).
As with, chisq, the degrees of freedom are
treated as a parameter to be estimated using (by default) the
link loge. However, the mean can also
be modelled with this family function.
See inv.chisqMeanlink
for specific details about this.
This family VGAM function handles multiple responses.