# When is further reduction than the order statistics not possible?

Let $$X_1, ..., X_n$$ be a random sample from a population with location pdf $$f(x-\theta)$$. Show that the order statistics $$T(X_1, ..., X_n) = (X_{(1)}, ..., X_{(n)})$$ are a sufficient statistic for $$\theta$$ and no further reduction is possible.

Thus far I know and understand as a statement that when we don't have much information about the parameters of a distribution as in a non-parametric case, then the order statistics are the maximum reduction that we can achieve. But how do I prove that the above statement holds true for $$f(x-\theta)$$. Letting $$Z_1, ..., Z_n$$ be defined as $$Z_i \equiv X_i - \theta$$ for all $$i=1,2,...,n$$, is it enough to show that the pdf $$f(x-\theta)$$ becomes $$f(z)$$, which provides no information about $$\theta$$?

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• This appears to be a question from a course or textbook. Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. – Silverfish Apr 12 '16 at 21:01
• Yes. This is from the book Statistical Inference by Casella and Berger. I am reading the book on my own to learn the subject (and not posting my homework.) Thus far I know and understand as a statement that when we dont have much information about the parameters of a distribution as in a non-parammetric case, then the order statistics are the maximum reduction that we can achieve. But how do i prove that the above statement holds true for f(x-θ). @Silverfish – user666 Apr 12 '16 at 21:16
• I suggest you edit that information into the question. Note that self-study questions get handled the same way regardless of whether it was homework for school or something you're teaching yourself (have a read of the wiki page I linked). You might also want to look at our editing help - you can do math typesetting here with Latex, e.g. writing $x$ produces $x$. – Silverfish Apr 12 '16 at 21:19
• Is it enough to show that considering the RV's Z1,..Zn such that Zi=Xi-θ (for all i=1,2,..n) the pdf f(x-θ) becomes f(z) which provides no information about θ? – user666 Apr 12 '16 at 21:19
• Now that the OP has added the tag & stated what they understand thus far, I believe this meets our criteria. This Q should be considered on topic here, IMO. – gung - Reinstate Monica Apr 12 '16 at 21:30

Since this is self-study -- Hint: Look at Fisher's Factorization Theorem and show that the likelihood for $\theta$ is a function of the order statistics.
You'll also need to show that it is minimal sufficient (so that no "reduction" is possible) by showing that the likelihood ratio for two samples is free of $\theta$ iff they have the same order statistics.