rotatingpaguro comments on Generalizing Koopman-Pitman-Darmois

rotatingpaguro 24 Dec 2023 20:29 UTC
3 points
In the “Independent But Non-Identical KPD” version, the term $\sum_{i \notin E} g_{i} (x_{i})$ is not a sufficient statistic in general, right?
(I could probably figure it out getting into the weeds of the proof, but I did not get it by reading once.)
- johnswentworth 24 Dec 2023 20:47 UTC
  3 points
  Parent
  Correct.
  The term $\sum_{i \notin E} g_{i} (x_{i})$ along with $x_{i}$ for all $i \in E$ is a sufficient statistic, of dimension at most 2D. So if we assume that the $X_{i}$ ’s are 1-dimensional, then there’s a “gap” in the theorem between D and 2D: there exists a summary statistic of dimension D only if the distribution has that form, and if the distribution has that form then $\sum_{i \notin E} g_{i} (x_{i})$ and $x_{i}$ for all $i \in E$ together form a summary statistic of dimension at-most 2D.
  - rotatingpaguro 24 Dec 2023 22:05 UTC
    3 points
    Parent
    I already understood that because you explain it in the text; the further doubt I have is: concerning only the “only if” part, given a $D$ -dimensional sufficient statistic exists by assumption, if $\sum_{i \notin E} g_{i} (x_{i})$ is also a $D$ -dimensional sufficient statistic or not.
    I think not, because it should not be able to capture what goes on with the $E$ variables, that’s hidden in the completely arbitrary $\prod_{i \in E} P [X_{i} | Θ]$ term.
    This annoys me because I can’t see the form of the sufficient statistic like in the i.i.d. case.