# Statistical Freedom

What is the English counterpart of the French term "statistique libre"?

The following excerpt, translated by me from an excellent 70's French textbook in mathematical statistics ([BAR]), defines the term "a free statistic" ("statistique libre"). As i am unfamiliar with either the term or its definition, and as i did not find a definition thereof in the French Wikipedia, i was wondering whether this concept has a standard English name, or whether it has not withstood the test of time and its use is essentially confined to this textbook.

Let $$\left(\Omega,\mathcal{A},\mathbf{P}\right)$$ be a statistical space.

1. An event $$A$$ is said to be free if $$P(A)$$ is the same value for all $$P\in\mathbf{P}$$.

2. A sub-$$\sigma$$-algebra $$\mathcal{B}\subseteq\mathcal{A}$$ is said to be free if each of its events is.

3. The statistic $$T$$, taking values in $$\left(\Gamma,\mathcal{B}\right)$$ is said to be free if $$T^{-1}\left(\mathcal{B}\right)$$ is, i.e. if $$T$$'s distribution is the same for all $$P\in\mathbf{P}$$.

-- Ch. II, Sec. 4, Definition 1

I should add that, as far as the author of the textbook is concerned, this is a very important concept: It features in the title of a chapter right next to "sufficiency" (chapter II: "Sufficiency and Freedom"), and the author goes on to write that the two notions of sufficiency and freedom are distinctive features of statistics that separate it from probability. (In the introduction to chapter II).

References

[BAR] Barra, Jean-René. Notion fondamentales de statistique mathématique. Dunod, 1971.

• I agree, +1. Commented Apr 25, 2013 at 6:45