# Is there a general expression for ancillary statistics in exponential families?

An i.i.d sample $$X1,…,XnX_1,\dots,X_n$$ from a scale family with c.d.f. $$F(xσ)F(\frac{x}{\sigma})$$ has $$S(X)S(X)$$ as an ancillary statistic if $$S(X)S(X)$$ depends on the sample only through $$X1Xn,⋯,Xn−1Xn\frac{X_1}{X_n},\cdots,\frac{X_{n-1}}{X_n}$$.

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. $$(ψ,χ)(\psi,\chi)$$ are the parameter of the exponential family, and then $$S=(T,A)S=(T,A)$$ is the minimal sufficient statistics for the parameter $$psipsi$$ of interest.We call the statistics $$AA$$ an S-cut if the distribution $$T∣AT\mid A$$ depends only on $$ψ\psi$$ and the distribution of $$AA$$ only depends on $$χ\chi$$.
If we look it via factorization, this actually prompt us to $$(T,A)(ψ,χ)d=(T∣A)ψ⋅(A)χ(T,A)_{(\psi,\chi)}\overset{d}{=}(T\mid A)_{\psi}\cdot (A)_{\chi}$$. Geometrically this only means that we can find a subspace for statistic $$AA$$ 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 ($$p†p^{\dagger}$$-formula, 6.10)for abritrary ancillary statistics and later proved it is $$√n\sqrt{n}$$-consistent. However, it does not reveal any geometric feature since if $$n→∞n\rightarrow\infty$$ 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.