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An i.i.d sample $X_1,\dots,X_n$ from a scale family with c.d.f. $F(\frac{x}{\sigma})$ has $S(X)$ as an ancillary statistic if $S(X)$ depends on the sample only through $\frac{X_1}{X_n},\cdots,\frac{X_{n-1}}{X_n}$.

  1. Is this result also sufficient?
  2. Is there a parallel result for a general exponential family which is not necessarily a scale family?(not asymptotic results, see update below)

If yes, I want to see some reference; If no, why is it not possible?

Note: Reading B. Efron's paper on the geometry on exponential families I now believe that this should somehow relate to the geometric nature of the exponential families.

But I have difficulty imagining what geometric object ancillary statistics should correspond to. Firstly I thought it should be the normal bundle, but later I found it only sufficient.


Update on this question:

After a careful look into [1], I think by ancillary statistics I mean the likelihood ratio (derived) ancillary statistics, not the Efron-Hinkley affine ancillary statistics. [1,Fig1] showed the difference in their marginal densities.

The results pointed out in [2] by a nice comment by @kjetilbhalvorsen below do not address my question.

[2] discussed several examples around pp.30-45, and proposed a simple case where S-sufficient S-ancillary are simultaneously introduced. i.e. $(\psi,\chi)$ are the parameter of the exponential family, and then $S=(T,A)$ is the minimal sufficient statistics for the parameter $psi$ of interest.We call the statistics $A$ an S-cut if the distribution $T\mid A$ depends only on $\psi$ and the distribution of $A$ only depends on $\chi$. If we look it via factorization, this actually prompt us to $(T,A)_{(\psi,\chi)}\overset{d}{=}(T\mid A)_{\psi}\cdot (A)_{\chi}$. Geometrically this only means that we can find a subspace for statistic $A$ and write them in form of direct product. This is not interesting since we know the minimal sufficient statistics does not always exist. One useful example in [2] is that it gave the approximation formula ($p^{\dagger}$-formula, 6.10)for abritrary ancillary statistics and later proved it is $\sqrt{n}$-consistent. However, it does not reveal any geometric feature since if $n\rightarrow\infty$ then any such aprroximation essentially describing locally Gaussian space.

[1]Pedersen, Bo V. "A comparison of the Efron-Hinkley ancillary and the likelihood ratio ancillary in a particular example." The Annals of Statistics (1981): 1328-1333.

[2]Cox, D. R., and O. E. Barndorff-Nielsen. Inference and asymptotics. Vol. 52. CRC Press, 1994.

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    $\begingroup$ Look at the books by Barndorff-Nielsen & Cox, the answer is in there, someplace $\endgroup$ Mar 19, 2017 at 20:41
  • $\begingroup$ @kjetilbhalvorsen Would you mind adding at least the publishing information of the title you are referring to? Thanks, $\endgroup$
    – Henry.L
    Mar 19, 2017 at 20:48
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    $\begingroup$ O.E.Barndorff-Nielsen&D.R.Cox: "Asymptotic Techniques for Use in Statistics" & "Inference and Asymptotics" (probably the last one) (Both books is Chapman&Hall) $\endgroup$ Mar 19, 2017 at 21:02
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    $\begingroup$ Ghosh, M., et al. “ANCILLARY STATISTICS: A REVIEW.” Statistica Sinica, vol. 20, no. 4, Institute of Statistical Science, Academia Sinica, 2010, pp. 1309–32, jstor.org/stable/24309506. $\endgroup$
    – krkeane
    Mar 29, 2022 at 13:51
  • $\begingroup$ Not yet. Exponential families can have multiple parameters and the distributions can vary widely. It's really hard to come up with a general expression that is independent of all these parameters. $\endgroup$
    – Lost Ninja
    Oct 14, 2023 at 7:45

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