Problem: Let $X_1, X_2, \dots , X_n$ be a random sample with densities $f_{X_i}(x) = exp(iθ − x)\mathbb{I}{x \geq iθ}$. Use Neyman’s Factorization Theorem to find a sufficient statistic for θ.

Solution: We have $\frac{x_i}{i} \geq \theta$, therefore $\min {X_1,\dots, \frac{X_i}{i}, \dots,\frac{X_n}{n}}$ is a sufficient statistic for $\theta$.