Show the maximum order statistic ${X}_{(n)}$ is not a sufficient statistic.

Question

Suppose $X_1,...X_n$ are independet random variables with $X_i \sim U( θ,θ+1)$ where θ is unknown. Show the maximum order statistic ${X}_{(n)}$ is not a sufficient statistic.

I know the pdf is: $f_\theta (x_i) = 1_{\theta<x_i<\theta+1}$.

Answer 1

Hint :

If you take a look at the joint distribution of your samples \begin{align*} f_\theta(x_1,\dots,x_n)&=\prod_{i=1}^n f_\theta(x_i)\\ &= \prod_{i=1}^n 1_{\theta<x_i<\theta+1}\\ &= 1_{\theta<x_{(1)}}1_{x_{{(n)}<\theta+1}} \end{align*} You can check that there is no way of writing this as a function of $x_{(n)}$ only.