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Suppose that $X_1,\cdots, X_n$ are independent random variables from uniform distribution on interval $(\theta_1,\theta_2)$, $\theta_2>\theta_1>0$. It is know that $T(X)=(X_{(1)},X_{(n)})$ is the complete sufficient statistic for $(\theta_1,\theta_2)$.

Find the expectations of the largest and smallest order statistics $X_{(n)}$ and $X_{(1)}$ respectively.

Hint: Use the transformation $Y=\frac{X-\theta_1}{\theta_2-\theta_1}$

How do I use the hint to solve the problem?

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  • $\begingroup$ You don't need to use the transformation. Hint: don't use the hint. $\endgroup$ – wolfies Mar 26 '15 at 6:07
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Hint: $Y$ is the normalized random variable to unit rectangle. In other words if $\hat{X_{(n)}}$ is the UMVUE of $\theta_1$ then $E[Y]=1$ for $X=\hat{X}_{(n)}$. Similarly, $E[Y]=0$ if $X=\hat{X}_{(1)}$

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$Y = (X-\theta_1)/(\theta_2-\theta_1)$ means that if $X\sim \mathcal U(\theta_1;\theta_2)$, then $Y\sim \mathcal U(0;1)$ and $\mathsf E(X_{(k)}) = \theta_1+(\theta_2-\theta_1)\mathsf E(Y_{(k)})$

Can you find $\mathsf E(Y_{(1)})$ and $\mathsf E(Y_{(n)})$ ?

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