[R-sig-ME] reference on unidentifiability of observation-level variance in Bernoulli variables?

[i white]

How about McCullagh and Nelder, Generalized Linear Models (2nd ed), 
section 4.5, particularly bottom of p125.

Kent Holsinger wrote:
> 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
>>
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>>
> 
> 

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