To address the questions regarding ancillary statistics in exponential families:

1. Sufficiency of Ratio-Based Ancillarity in Scale Families

For an i.i.d. sample from a scale family with c.d.f. ( F\left(\frac{x}{\sigma}\right) ), a statistic ( S(X) ) that depends only on the ratios ( \frac{X_1}{X_n}, \ldots, \frac{X_{n-1}}{X_n} ) is indeed ancillary (its distribution is independent of...

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...

1. EXACT SCALE FAMILIES

Let

    X = (X1 , … , Xn) ,           Xi  i.i.d.  with c.d.f.  F(x/σ) ,  σ>0

and let G = (0,∞) be the multiplicative group acting on the sample space ℝn by

    g·x = (g x1 , … , g xn)   (g>0 , x∈ℝn).

The model is equivariant: Pσ is the image of P1 by the action g = σ.
Put

    I(x) = ( x1/xn , … , x(n-1)/xn)

(any other complete s...

This is an interesting question about the structure of ancillary statistics in exponential families. Let me address your questions systematically.

1. Sufficiency of the Scale Family Result

For scale families with CDF , you correctly note that functions of the ratios are ancillary. This is indeed necessary and sufficient for scale families.

**Pr...

<think> We are addressing a question about ancillary statistics in exponential families. The original question has two parts:

1. Is the result that for a scale family with c.d.f. ( F(x/\sigma) ), an ancillary statistic ( S(X) ) depends on the sample only through the ratios ( X_1/X_n, \dots, X_{n-1}/X_n ) also sufficient? That is, is the converse true?

2. Is there a parallel result for a ge...

Ancillary Statistics in Exponential Families: Geometric Perspective

To properly address your question about ancillary statistics in exponential families, I'll need to clarify some concepts and then address your specific queries.

For Scale Families

First, regarding your question about scale families: Yes, the condition is both necessary and sufficient. If a statistic S(X) depends on the sam...

This is an excellent and deep question that touches upon some foundational concepts in statistical inference and their geometric interpretations. Let's break it down.

1. Ancillary Statistics in Scale Families

You are correct about the characterization of ancillary statistics for a pure scale family. Let's formalize it to answer your first question.

Let be an i.i.d. sample ...

Below the notation

X = (X₁,…,Xₙ) sample of size n
Pσ law of X when the density is (1/σ)f(x/σ) (scale family)
T(X) a (minimal) sufficient statistic
A(X) a candidate ancillary statistic (its law must not depend on the parameter)

1. The scale family

The model {Pσ : σ>0 } is generated by the action of the multiplicative group G = ℝ⁺ on the sa...

Here is what one finds if one works it out carefully:

1. Scale‐families (and, by the same token, any one‐dimensional group–family)
–– If (X_1,\dots,X_n) are i.i.d. from a scale family with c.d.f.\ (F(x/\sigma)), then the σ–field of all ancillary statistics is exactly the σ–field generated by the maximal invariant under the scale–group, namely [ \Bigl{\tfrac{X_1}{X_n},,\do...