When we define Fisher information as the variance of the score function $ \frac{\partial \log f(X|\theta)}{\partial \theta}$, are we assuming the regularity conditions?
Obviously we need the regularity conditions to prove that $\mathbb{E}[ \frac{\partial \log f(X|\theta)}{\partial \theta}] = 0$ and $I(\theta) = \mathbb{E}[- \frac{\partial^2 \log f(X|\theta)}{\partial \theta^2}] $ as well, but what about the actual definition itself?
In that case, can we say that the Fisher information of uniform distribution $\mathit{U}(0,\theta)$ is $I(\theta) = \mathbb{E}[(\frac{\partial \log f(X|\theta)}{\partial \theta})^2] = \frac{n}{\theta^2}$.
