reference on unidentifiability of observation-level variance in Bernoulli variables?

Ben,

Here's how I think about it. (I know this isn't printed anywhere, but 
maybe it will be good enough for your purposes.)

Suppose we put a beta prior on p, Be(alpha*pi, alpha*(1-pi)), where 
alpha = (1-theta)/theta. Then the mean of p is pi and its variance is 
theta*pi*(1-pi). If we have only one observation, we can't estimate both 
pi and theta. The same argument would hold for *any* continuous 
distribution that has more than one parameter, i.e., any non-degenerate 
distribution.

Kent

On 11/15/11 3:53 PM, Ben Bolker wrote:

   Hi folks,

   It has often been discussed on this list and other R-help lists that
overdispersion is (broadly speaking) unidentifiable for an ungrouped
Bernoulli response variable (if some kind of grouping can be imposed,
then it becomes identifiable). For example:


https://stat.ethz.ch/pipermail/r-help/2008-February/154058.html
http://finzi.psych.upenn.edu/R/Rhelp02a/archive/91242.html

Peter Dalgaard:

The point being that you  cannot have a distribution on {0, 1} where>

the variance is anything but  p(1-p) where p is the mean; if you put>  a
distribution on p and integrate  it out, you still end up with the>
same variance.

    I am curious if anyone has a *printed* (book or peer-reviewed
article) for this, or even can point to notes that actually go to the
trouble of doing the integration and proving the statement.  I have
looked in some of the usual places (MASS; McCullough, Searle, and
Neuhaus; Zuur et al.) and haven't come across anything.

   Something that showed the computation of the intraclass correlation
and worked out the mean of the logistic-normal-binomial distribution as
a function of the mean and variance of the underlying normal
distribution would be nice too, although I'm guessing it doesn't have a
straightforward analytical solution ...

   I'm hoping I haven't missed anything obvious -- on the other hand, if
I have it will be easy for someone to answer (and please don't be
offended if it was in your book, which was on my shelf all the time and
I forgot to look there ...)

  thanks
   Ben Bolker