Long time ago (early 1980's) my professor showed me a paper (I think it was Teachers' Corner or something similar) about the Poisson distribution and completeness. Showed that if only one point was removed from the parameter space, then the usual sufficient and complete statistic was no longer complete! Now, using to much time googling and on JSTOR, I am unable to find such a paper. Anybody has any idea?

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    $\begingroup$ I wonder if it might have popped up in a Letter to the Editor -- those tend to be harder to locate. $\endgroup$ – Glen_b Feb 25 '19 at 6:33

I'm starting to think that paper must/may have been retreated, because the result (at least as I remember & cited it) cannot be true. That is because it is contradicted by theorem 2.12 of Brown: Essentials of Statistical Exponential Families ungated link. This theorem says: Let $\{p_\theta \colon \theta \in \Theta\}$ be a standard exponential family with $\Theta^\mathrm{o}\not=\emptyset$, the interior (in topologic sense) of the parameter space must be nonempty. Then the family $\{p_\theta\}$ is complete.

The poisson family is a exponential family, and the natural parameter space is the positive real line. Removing only one point still leaves an open interior.


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