Let $X_1,\ldots,X_n$ be a random sample from $U(\theta-\frac{1}{2},\theta+\frac{1}{2})$. Show that the statistics $T(X_1,\ldots,X_n)=(X_{(1)},X_{(n)})$ is a sufficient statistics for $\theta$.

Can someone show me how to use the theorem below in proving the above question:

Theorem If $p(x|\theta)$ is the joint pdf or pmf of $X$ and $q(t|\theta)$ is the pdf or pmf of $T(X)$, then $T(X)$ is a sufficient statistic for $\theta$ if, for every $x$ in the sample space, the ratio $$\frac{p(x|\theta)}{q(T(x)|\theta)}$$ is a constant as a function of $\theta$.

  • $\begingroup$ What's wrong with using Factorization theorem? $$f_{\theta}(x_1,\ldots,x_n)=\mathbf1_{\theta-1/2<x_{(1)},x_{(n)}<\theta+1/2}=\mathbf1_{x_{(n)}-1/2<\theta<x_{(1)}+1/2}\quad,\,\theta\in\mathbb R$$ $\endgroup$ – StubbornAtom Feb 5 at 20:30
  • $\begingroup$ @StubbornAtom, there is absolutely nothing wrong. I just want to see if the above can also be used in proving. $\endgroup$ – Lady Feb 5 at 20:34

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