Short answer

1. For an i.i.d. scale model X1,…,Xn with distribution F(x/σ), the “ratio” characterization is not only sufficient but (modulo null sets and under standard regularity) also essentially necessary: an exact ancillary is, almost surely, a function of a maximal invariant under the scale group. Vectors such as (X1/Xn,…,Xn−1/Xn), or any equivalent maximal invariant (e.g. X/∥X∥ for a chosen norm, ratios of order statistics), generate the ancillary σ-field. Thus, S is ancillary if and only if S is measurable with respect to the σ-field generated by a maximal scale-invariant.

2. There is no comparable general characterization for arbitrary exponential families. Exact nontrivial ancillaries beyond the classical transformation (group-invariant) models are rare; when they do exist they typically arise because the family is a transformation model (location, scale, location–scale, rotation, etc.), in which case the maximal invariant is ancillary. In a general (regular, full) exponential family there is no universal expression for ancillaries; most of the useful “ancillaries” that appear in practice are approximate (likelihood-ratio or affine ancillaries) with asymptotic justification.

Details and references

• Scale families as transformation models:

  • Let G = (0,∞) act on the sample space by g·x = gx. The family is a G-transformation model: Pσ = P1 ∘ gσ−1. Any G-invariant statistic has distribution free of σ, hence is ancillary (sufficiency).
  • Conversely, let S be ancillary. Then for all σ, under P1 the distributions of S and S ∘ gσ coincide. Under mild regularity, this implies S is measurable with respect to the invariant σ-field and, hence, is a function (a.s.) of a maximal invariant such as the vector of ratios. In other words, the invariant σ-field is a maximal ancillary σ-field.
  • References:
    • T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed., Sect. 10. (discussion of maximal invariants in scale/orthogonal groups)
    • D. M. Wijsman, Invariant Measures on Groups and Their Use in Statistics, Chs. 1–2.
    • M. L. Eaton, Group Invariance in Statistics, esp. Ch. 2 (Ancillarity and invariant σ-fields); see Theorem 2.2-type results stating that invariant σ-fields are ancillary and maximal among ancillary σ-fields in transformation models.
    • E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses (3rd ed.), Sect. 6.6 (Invariance, maximal invariants, and ancillarity).

• General exponential families:

  • A regular full exponential family with density exp{η(θ)T(x) − ψ(θ)}h(x) has T(x) minimal sufficient for θ. The conditional distribution of X given T = t is θ-free (by sufficiency), so any function of X measurable given T has a θ-free conditional distribution. However, it becomes marginally ancillary only if it is also independent of T for all θ, which typically requires a genuine transformation symmetry. Absent such a group structure, exact nontrivial ancillaries generally do not exist and there is no general form for them.
  • This is why exact ancillaries are standard in transformation families (location, scale, rotation) via maximal invariants, but otherwise one uses approximate ancillaries (likelihood-ratio or affine) for higher-order conditional inference.
  • References:
    • D. R. Cox and O. E. Barndorff-Nielsen, Inference and Asymptotics, Chs. 3, 6 (conditional inference, ancillarity; emphasizes rarity of exact ancillaries outside transformation models).
    • N. Reid, The roles of conditioning in inference, Statistical Science 10 (1995), 138–199 (survey; exact ancillaries mostly in transformation models; otherwise approximate ancillaries).
    • D. A. S. Fraser, The Structure of Inference (1968), Chs. 3–4 (ancillarity and transformation models; maximal invariants as ancillaries).
    • M. L. Eaton, Group Invariance in Statistics (as above), for the precise group-theoretic framework that yields exact ancillaries.

Notes on your update (likelihood-ratio ancillaries)

• The likelihood-ratio derived ancillary used in higher-order inference is, in general, an approximate ancillary: geometrically it follows the observed likelihood contours through the MLE and is orthogonal (in an information-geometry sense) to the parameter of interest. It coincides with exact ancillaries in transformation models but is only asymptotically ancillary otherwise. This is why Barndorff-Nielsen–Cox and Efron–Hinkley constructions are powerful but inherently asymptotic unless a group structure is present.

Takeaways

• For scale families, ancillarity is exactly characterized by scale invariance: S is ancillary iff it is a function of a maximal scale-invariant (e.g., ratios).
• For general exponential families, there is no general exact counterpart; exact ancillaries are essentially a feature of transformation (group-invariant) models, and otherwise one relies on asymptotic (likelihood-based) ancillary constructions.