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 $$E(X_{(n)})=\frac{(2n+1 )\theta}{n+1}$$

Thus,

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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