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

It is known that 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: I thought a bit about this question for a few days, then I ended up reading B. Efron's paper on the geometry on exponential families and now I believe that this should somehow related 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 , 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  by a nice comment by @kjetilbhalvorsen below do not address my question.

 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  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.

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.

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

• Look at the books by Barndorff-Nielsen & Cox, the answer is in there, someplace – kjetil b halvorsen Mar 19 '17 at 20:41
• @kjetilbhalvorsen Would you mind adding at least the publishing information of the title you are referring to? Thanks, – Henry.L Mar 19 '17 at 20:48
• 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) – kjetil b halvorsen Mar 19 '17 at 21:02