Complete Sufficient Statistic of Uniform $(\theta,2\theta)$

by GAGA   Last Updated July 12, 2019 06:19 AM

Let $X_1,....,X_n$ be iid Uniform $(\theta,2\theta)$ , $\theta >0 $

It is easy to show that $T=(X_{(1)}, X_{(n)} )$ is a sufficient statistic for $\theta $ and we want to show it is not a complete sufficient statistic.

So my goal is to:

find a non-zero function $g(T)$ for which $E(g(T))=0$ for all values of θ.

I calculated : $E(X_{(1)})=\frac{(n+2 )\theta}{n+1}$ and $E(X_{(n)})=\frac{(2n+1 )\theta}{n+1}$


Step 1: we can choose as $g(T)$ $= (n+2)X_{(n)} - (2n+1) X_{(1)} $ , How can we prove with words that this is a non zero function?

Step 2 : But $E(g(T))= $E$[$$(n+2)X_{(n)} - (2n+1) X_{(1)}$$]$= By linearity of expecatation

$E$[$(n+2)X_{(n)}$] $-$ $E$[$(2n+1) X_{(1)}$]= $0$

Hence from Step 1 and Step 2 we can conclude that $T=(X_{(1)}, X_{(n)} )$ is not complete.

Is this correct? also what else can I write in step 1 to prove that it is non zero function

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