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

An i.i.d sample X1,,Xn from a scale family with c.d.f. F(xσ) has S(X) as an ancillary statistic if S(X) depends on the sample only through X1Xn,,Xn1Xn.

  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) 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 TA depends only on ψ and the distribution of A only depends on χ.
If we look it via factorization, this actually prompt us to (T,A)(ψ,χ)d=(TA)ψ(A)χ. 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-formula, 6.10)for abritrary ancillary statistics and later proved it is n-consistent. However, it does not reveal any geometric feature since if n 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.


Source : Link , Question Author : Henry.L , Answer Author : Community

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