 Solution

The score function isWith common parameter-independent support and justified differentiation under the integral, ${\mathbb{E}}_{\theta }[{\partial }_{\theta }\log f(X_1,\theta )]=\int {\partial }_{\theta }f(x,\theta )dx={\partial }_{\theta }1=0$. Linearity then gives $\boxed{{\mathbb{E}}_{{\theta }_0}{S}_n({\theta }_0)=0}$. These regularity conditions exclude, for example, moving-support uniform models.