I am not able to find the minimal sufficient statistic for the following density function:
$$f(x_i;\theta) = 2(1+\theta-x_i)I_{\theta \le x_i \le \theta+1}$$
The function does not belong to the exponential family distribution and so I apply the the Lehmann Scheffé Theorem, from which I get that a minimal sufficient statistic should be:
$$T(\mathbf x) = (\min(x_i), \max(x_i), \prod(1+\theta+x_i))$$
But since a statistic cannot depend on the parameter, it is wrong for sure.