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 $\frac{X_1}{X_n},\cdots,\frac{X_{n-1}}{X_n}$.
- Is this result also sufficient?
- 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.
