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Let $X_1,X_2,X_3$ be iid random variables such that $E(X_1)=\mu$

Define $X_{(3)}$ and $X_{(1)}$ as the maximum and minimum order statistics respectively.

I know that if $X$ is normal, $R=X_{(3)}-X_{(1)}$ is ancillary to $\mu$

What is the biggest family of distributions for which this is true?

kjetil b halvorsen
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Marj
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    Could you tell us how "biggest" is supposed to be determined? Cardinality, number of parameters, set inclusion, or maybe something else? – whuber Dec 02 '19 at 20:26
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    This would be true for symmetric distributions, but maybe also some others ... – kjetil b halvorsen Dec 02 '19 at 23:24
  • Thank you. I guess biggest is somewhat vague. I was I meant biggest as in most general. Symmetric is pretty broad which helps a lot. – Marj Dec 03 '19 at 15:34

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This will be true for all symmetric distributions. But as the linked post makes clear, distributions might be symmetric in different senses, and you didn't specify which. But in statistics, the most used sense is symmetry with respect so some point of symmetry, usually the expectation (when exists.) So I will assume that definition.

So, the result will hold for the family of all symmetric distributions (with existing expectation.) But: Let $X \sim F$ for some distribution $F$ (with existing expectation. Will not repeat that.) Then define a distribution family by $ Y = X+\theta; X \sim F$ for $\theta \in \mathbf{R}$. Then obviously the distribution of $R$ will be free of $\theta$, so will be ancillary for the expectation. One could maybe/probably construct more such families. And this families will not be ordered by inclusion ... so it is not possible to tell which is largest (in the sense of set inclusion.) Now you can understand the comment by @whuber.

kjetil b halvorsen
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    The correctness of this answer depends on what definition of "ancillary" you are using. As you point out, any given *location family* generated by a *single* symmetric distribution will work. But consider, say, the Normal$(\mu,\sigma)$ family: the range *does* give information about its parameter and--according to [this definition](https://en.wikipedia.org/wiki/Ancillary_statistic)--is *not* ancillary. Accordingly, the family of all symmetric distributions that you name at the outset doesn't work. – whuber Dec 03 '19 at 00:08
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    @whuber: The question says *ancillary to $\mu$*, that is the sense I am using. This corresponds to the concept of *S-nonformation* given on page 47 of [Barndorff-Nielsen](https://www.amazon.com/Information-exponential-families-statistical-mathematical/dp/0471995452/ref=sr_1_12?keywords=barndorff-Nielsen&qid=1575332593&sr=8-12) which I have also linked [here](https://stats.stackexchange.com/questions/410867/why-arent-error-in-x-models-more-widely-used/412772#412772). – kjetil b halvorsen Dec 03 '19 at 00:25
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    Thanks for the clarification, +1. – whuber Dec 03 '19 at 00:47
  • Any density $f$ such that$$\int xf(x)\text{d}x=0$$leads to a family$$f_\mu(x)=f(x-\mu)$$that makes $R$ ancillary in $\mu$. – Xi'an Dec 03 '19 at 05:01
  • @Xi'an: That is included in my answer ... – kjetil b halvorsen Dec 03 '19 at 05:05