[R-sig-ME] question on nbinom1

[Don Cohen]

In response to a question about use of nbinom1 in glmTMB, specifically
about this paper by VerHoef J.M. & Boveng:
 https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1141&context=usdeptcommercepub
which says that AIC cannot be used to choose between a quasi-Poisson model
and a negative binomial,

Ben Bolker writes:

 >     nbinom1 is *not* quasi-: see Hardin & Hilbe 2007 as suggested in 
 > ?nbinom1.  (It can't be fitted in a standard GLM framework, since this 
 > parameterization of the nbinom doesn't fit into the exponential family, 
 > but glmmTMB is more flexible than that ...)

I've been trying to figure out what Hardin & Hilbe have to say about this,
and so far failing to find the connection.
(BTW, is nbinom1 the same as what they call NB-C ?)

Does "nbinom1 is *not* quasi-" mean that the AIC reported when I use 
nbinom1 really IS comparable to the one reported when I use nbinom2?
And that loglik is actually being computed for a "real" distribution?
Does it matter whether I also use random effects, zero inflation, 
offset, etc. ?

Alternatively, if AIC for nbinom1 is really NOT comparable to AIC for the
other (real?) distributions, how is one supposed to determine which model
better fits the data?  VerHoef suggests trying to determine whether the
variance for different subpopulations with different means more closely
resembles linear or quadratic functions of the mean, but this is not at
all clear in the data I'm trying to analyze.