Let $X_1, X_2, ..., X_n$ be $n$ independent random variables, where $X_i$~uniform($i\theta$), $i=1,2,...,n$. Find a sufficient statistics for $\theta$.
My Attempt
The conditional distribution :
$$ f( x_1 ,\ldots ,x_n \mid i\theta ) = \frac 1 {(i\theta)^n}, \text{ where } ( x_i \leq i\theta , i=1,2,\ldots,n).$$
Rewriting the conditional distribution as:
$$ f( x_1 ,\ldots ,x_n \mid i\theta ) = {(i\theta)^{-n}} \quad ( x_i \leq i\theta , i=1,2,\ldots,n) \tag 1$$
So,
$$ f( x_1 ,\ldots ,x_n \mid \theta ) = {(i\theta)^{-n}} \quad ( \max(x_i) \leq i\theta)\tag 1$$
Thus, the sufficient statistic $\max(x_i/i)%$ is taken.
My question is, how to I take into account uniform($i\theta$) for the discrete case as the question implies?