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Is statistic $\max\{-X_{(1)},X_{(n)}\}$ one dimension or two dimension?

I was trying to find the minimal sufficient statistic for $U(-\theta,\theta)$ from $n$ $i.i.d$ random variables $X_i$. The result is that $\theta\ge \max\{-X_{(1)},X_{(n)}\}$ thus the minimal sufficient statistic is $\max\{-X_{(1)},X_{(n)}\}$. However, the problem actually states that "Find a two dimensional minimal sufficient statistic for $U(-\theta,\theta)$". Is $\max\{-X_{(1)},X_{(n)}\}$ a two dimensional statistic?

Here $X_{(i)}$ is the $i^{th}$ smallest value of $X_1,\cdots,X_n$.

Tan
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1 Answers1

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I presume by $\{X_{(i)}\}$ you mean the ordered sequence constructed from $\{X_i\}$. The minimal sufficient statistics is the pair of $(min\{X_i\}, max\{X_i\})$, however, if the sequence is ordered, then you are right; the minimal sufficient statistics is $max\{-X_{(1)}, X_{(n)}\}$, but it requires an additional step of ordering.

Correction

Yes. You are right. $max\{|X_i|\}$ is the sufficient statistics.

Arash
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  • I think the minimal sufficient statistic of $\theta$ using ${X_i}{i=1}^n$ is $\max{-X{(1)},X_{(n)}}$, not $(X_{(1)},X_{(n)})$. – Tan Jan 04 '21 at 22:51
  • If ${X_i}$ is already increasingly ordered, yes you are right. But if not, then you are going to have to find both min and max of the sequence, hence a pair. – Arash Jan 04 '21 at 22:54
  • https://math.stackexchange.com/questions/1973864/minimal-sufficient-statistics-for-uniform-distribution-on-theta-theta – Tan Jan 04 '21 at 22:56