Is it okay to use an indicator function and have two statistics in the support? And, if this is possible, would the indicator function be a function of both statistics?
We have that $\mathbf{X}$ is a random sample from Uniform$(\theta, \theta+1)$ and we want to re-write the joint pdf from $$f(\mathbf{x}| \theta) = \prod \mathbf{1} [\theta < x_i < \theta +1]$$ to this new pdf $$f(\mathbf{x}|\theta) = \mathbf{1} [\max(\mathbf{x}) - 1 < \theta < \min(\mathbf{x})]$$ Does this new function help us find sufficient statistics or minimal sufficient statistics? Is it okay to treat indicator functions this way?