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I'm going through Statistical Inference by Casella and Berger, and I'm currently on Chapter 6. In particular, I'm doing exercise 6.9 (c), and I'm trying to prove that if $X_1, ..., X_n$ is a random sample from a population with pdf $f(x|\theta)=\frac{e^{-(x-\theta)}}{(1+e^{-(x-\theta)})^2}$, then the order statistic $T(X) = (X_{(1)},...,X_{(n)})$ is a minimal sufficient statistic for $\theta$.

The joint pdf of the sample is: $$f_X(\boldsymbol{x}|\theta)=e^{n(\theta-\bar{x})}\prod_{i=1}^n \frac{1}{(1+e^{-(x_{(i)}-\theta)})^2}.$$

I want to apply Theorem 6.2.13, so I want to verify that for every two sample points $\boldsymbol{x}$ and $\boldsymbol{y}$, the ratio $\frac{f(\boldsymbol{x}|\theta)}{f(\boldsymbol{y}|\theta)}$ is independent of $\theta$ if and only if $T(\boldsymbol{x}) = T(\boldsymbol{y})$. This ratio is:

$$e^{n(\bar{y}-\bar{x})} \prod_{i=1}^n \bigg(\frac{1+e^{-(y_{(i)}-\theta)}}{1+e^{-(x_{(i)}-\theta)}}\bigg)^2.$$

Now the first part is independent of $\theta$, so we just need to verify (dropping the square) that $$\prod_{i=1}^n \frac{1+e^{-(y_{(i)}-\theta)}}{1+e^{-(x_{(i)}-\theta)}}$$ is independent of $\theta$ if and only iff the $T(\boldsymbol{x})=T(\boldsymbol{y})$ are equal. In the solutions and everywhere else I've looked online this is just stated as true, and while the "if" direction is obvious, the "only if" one isn't. Or rather, sure, it looks like this should only be independent of $\theta$ if the order statistics are equal and I can't find a counterexample, but it doesn't seem necessarily obvious to me. I do believe it's true, but I can't prove it, and I can't even prove this for the case $n=2$, where the relevant part is

$$\frac{1+e^{\theta}(e^{-y_{(1)}}+e^{-y_{(2)}}) + e^{2\theta}e^{-y_{(1)}-y_{(2)}}}{1+e^{\theta}(e^{-x_{(1)}}+e^{-x_{(2)}}) + e^{2\theta}e^{-x_{(1)}-x_{(2)}}}.$$

I'm not sure if I'm missing something obvious, but any help with rigorously proving this would be appreciated (even if you can just answer the question for the special case $n=2$).

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It is more evident that it holds if you rewrite:

$$\prod_{i = 1}^{n} \frac{(1 + e^{-(y_{i} - \theta)})^{2}}{(1 + e^{-(x_{i} - \theta)})^{2}} = \prod_{i = 1}^{n} \frac{(e^{-\theta} + e^{-y_{i}})^{2}}{(e^{-\theta} + e^{-x_{i}})^{2}}$$

In this case, it is immediate to show that the expression becomes independent from $\theta$ (that is, all the $e^{-\theta}$ gets cancelled) if and only if for all $x_{i}$ exists a corresponding $y_{j}$ such that $x_{i}$ = $y_{j}$. Indeed, it is sufficient the existence of even one only $x_{i}$ for which there's not an identical $y_{j}$ that you'll end up having:

$$\frac{(e^{-\theta} + e^{-y_{j}})^{2}}{(e^{-\theta} + e^{-x_{i}})^{2}}$$

which can't be simplified to be independent from $e^{-\theta}$.

In other words, to hold, the two samples have to be one a $permutation$ of the other, something like, for instance, with $n$ = 4, $\boldsymbol{x}$ = $(3, 7, 4, 22)$, $\boldsymbol{y}$ = $(7, 22, 3, 4)$, and this clearly implies that also the order statistics have to be equal, so that $T(x) = (x_{1},...,x_{n}) = (y_{1},...,y_{n}) = T(y)$, proving the claim.

EDITING TO REPLY COMMENT

To prove the "only if" you need to assume $$A = \prod_{i = 1}^{n} \frac{(1 - e^{-(y_{i} - \theta)})^{2}}{(1 - e^{-(x_{i} - \theta)})^{2}}$$ being independent from $\theta$, which is never the case unless the two samples are one permutation of the other (previous part of my reply). In words, one sample being permutation of the other is itself necessary and sufficient condition for $A$ being independent from $\theta$, that is for the "only if" assumption to hold. From that, then follows the order statistics being minimal sufficient, which is indeed your "only if" statement: $$A \bot \theta \Rightarrow T(x) = T(y)$$

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  • $\begingroup$ Thanks for the reply, but you just restated what I wrote in my original post. The crux of the issue is that the "only if" part of "if and only if" is not immediate or trivial. It's not really obvious that only the permutations work (even though it is obvious that they themselves do work), and my question was how to prove rigorously that this is the case. $\endgroup$ – Ryker Apr 3 at 2:15
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    $\begingroup$ I saw your edit, but "which is never the case unless the two samples are one permutation of the other" is something that I acknowledged in my post already, but I was asking for proof of it. This is not something that can be just stated, but thankfully I found a proof while looking for something else. In case you wish to see what I was looking for, see Example 1.2.23 in stat.colostate.edu/~riczw/teach/STAT730_S15/Lecture/…, and the answer here: math.stackexchange.com/questions/2975830/…. $\endgroup$ – Ryker Apr 5 at 15:01

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