Re: Godambe's paradox

On 2 Sep 2003 07:40:30 -0700, [EMAIL PROTECTED] (Robert Dodier)
wrote:

>Joseph K. <[EMAIL PROTECTED]> wrote:
>
>> Does anyone have ideas regarding the meaning/resolution
>> of Godambe's paradox?
>>
>> Godambe VP. 1982. Ancillarity principle and a statistical paradox.
>> J. Am. Stat. Assoc. 77:931-933.
>
>I read the article, but I'm not seeing what the fuss is about;
>I must be missing something. Maybe someone can fill me in.
>
>Here is a summary. Interested readers are referred to the original.
>
>Let 2N slips of paper be put on a table. On the reverse of each
>slip is +1 or -1. N slips have +1 and N have -1. The state of
>nature is the arrangement of +1 and -1 among the slips; there
>are (2N choose N) states. Of the 2N slips, N are selected at
>random. What can we say about the arrangement of +1 and -1?
>
>At this point, Godambe asserts that any arrangement which
>assigns any proportion of +1 substantially greater or less
>than 1/2 is "implausible". Yet, he says, the distribution
>of the sample is not a function of the arrangement. The second
>statement makes the observed sample an ancillary statistic,
>but the first makes it possible (apparently) to make some
>inference; one is not supposed to be able to extract
>information from an ancillary statistic, so here is the paradox.
>
>I'm not seeing any paradox here. Any arrangement which does not
>assign exactly the observed sequence of +1 and -1 is not just
>"implausible", but impossible.

What you miss Robert is that Godambe was able to make "the usual
inference" (that is, the usual frequentist inference) without actually
seeing the values of +1s and -1s, which remained unknown. He was able
to make the "usual inference" by using as the sample only the labels
of the population units, {i; i=1,...,n}. Just after 2.6 in the paper
he wrote: "Note. In the preceding mode of inference the observation
(or data) consists of s' only. In particular (tetha_i:i belongs to s')
is not part of the data".

>So it is not true that
>the distribution of the sample is independent of the arrangement;

It is independent of the sequence of tethas that compose the parameter
since the actual values are not seen, only the labels. We have here a
case where a function of the sample is independent of the parameter to
be estimated (i.e. is the same for all tetha, iow, the function is
ancillary) and paradoxically the values of tetha which are plausible
and implausible can be identified (i.e. tetha can be estimated), using
the distribution of the function. 

The first version of the paradox appeared as a comment on a paper of
Dawid (1979, Conditional independence in statistical theory. J. R.
Stat. Soc. 41:1-15). Dawid replied rather poorly (imo), then Good
(1980. On Godambe's paradox. J. Statistical Computation and Simulation
12:70-72) stated "Godambe (1979) proposed a paradox which, if left
unresolved, would destroy the entire foundations of statistics,
Bayesian and non-Bayesian", and attempted to dissolve the paradox
without much success (imo) by arguing that in fact the function
depended on tetha. Godambe refined his case and published the full
account of the paradox in the paper quoted in my first post, including
a new and pretty complete and complex example using balanced sampling.
Genest and Schervish (1985, Resolution of Godambe's paradox, Can. J.
Stat 13:293-297) made a similar attempt at dissolving the paradox by
simply saying that the first example (of the paper slips described by
Robert) in Godambe (1982) is a "psychological trap" and in the second
example the function is not ancillary but actually depended on tetha.
Fraser commented on Genest and Schervish resolution and found it
faulty and Godambe himself found the argument absurd. Finally, in an
interview in 1988: 
(Godambe VP. 1988. A Conversation with V.P. Godambe. Liaison 2, No. 3.
(http://www.ssc.ca/main/about/history/godambe_e.html).
he said:
Godambe: "Yes. When the analysis of the role of randomization was
carried to its logical conclusion, I could see that there was a
paradox involved. Apparently, using the same arguments in a very
elementary situation, one is making inference about an unknown
constant exclusively on the basis of the realized value of a random
variate and its completely specified distribution. That is,
mathematically the distribution is independent of the unknown
constant. That was discussed in The Canadian Journal, and in JASA,
JRSS and elsewhere (7, 8, 9). And I would like people to discuss it
more because I myself do not clearly see the solution. I often think
that just as Russell’s paradox ultimately was resolved in terms of
analysis of language, similarly here, by properly restricting the
definition of parameter, we could eliminate this one. But this is just
a very obscure kind of feeling I have."

7 and 8 have been given above.
9. Bhave, S.V. ; Statist. Prob. Letters, 5 (1987), 243-246.

Joseph K.