Sufficient statistic with...

Question

Let be $X_1,X_2,..X_n$ independent random variables such that $f_k(x;\theta)=e^{k\theta-x}$ for $x>k\theta$. The problem is to find a sufficient statistic for $\theta$. I don't achieve to replace $I_{(n\theta,\infty)}(x_i)$ (that is 1 when $x_i>i\theta$ $\forall{i}\in\{1,2,...,n\}$ and $0$ in any other case) from $f(x_1,x_2,...;x_n;\theta)=exp\{{\theta\frac{n(n+1)}{2}-\displaystyle\sum_{i=1}^{n}}x_i\}I_{(i\theta,\infty)}(x_i)$, for use the factorization theorem

Answer 1

Note that the condition $x_i>i\theta$ is equivalent to ${x}_{i}/i>\theta$ then you can replace the indicator function with this last and take as suficient statistic the $min({x}_{i}/i)$