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From a note by Jun Shao

If $V(X)$ is a nontrivial ancillary statistic, then $σ(V(X)) ⊂ σ(X)$ is a nontrivial σ-field that does not contain any information about $P$.

I was wondering in what sense "the σ-field of statistic $V(X)$ does not contain any information about the distribution $P$ of $X$" means?

  1. I learned that σ-field itself represents information, but not sure about what it means by σ-field of $V(X)$ not containing information about distribution of $X$.
  2. Does "information" here mean the fisher information?

Thanks and regards

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It means that the probability distribution of V does not involve the parameter, or, if you prefer, that the Fisher information is null.

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