proof of Cramer-Rao lower bound

I am trying to understand the proof for this theorem from the book Casella and Berger (2nd ed.) page 336. $$W(X)$$ if any estimator for samples $$X_1,\ldots,X_n$$ on distribution $$f(X|\theta)$$. I notice that there is no assumption of unbiasedness for $$W(X)$$ till equation 7.3.8 which reads $$E_{\theta}\left(\frac{\partial}{\partial\theta}log f(X|\theta)\right)=\frac{d}{d\theta}E_{\theta}=0$$ This makes sense according to the argument given in the book where they substitute $$W(X)=1$$ to come to the above result. But I fail to see this equation holding in general. More specifically, if I think of a uniform distribution between 0 and $$\theta$$ (where $$\theta$$ parameterize the distribution), then $$f(x|\theta)=\frac{1}{\theta}$$. And hence, $$E_{\theta}\left(\frac{\partial}{\partial\theta}log f(X|\theta)\right)=\int_{\mathcal{X}}\frac{\partial}{\partial{\theta}}log f(x|\theta)f(x|\theta)dx=\int_{\mathcal{X}}\frac{\partial}{\partial{\theta}}f(x|\theta)dx=-\int_{0}^{\theta}\frac{dx}{\theta^2}\neq0$$

What am I not understanding?

edit: After πr8 suggestion, I tried it on exponential and normal distribution and it does give 0. Is discontinuity at boundary of the example I gave the reason?

• there will be extra smoothness assumptions somewhere - which conditions do they place on $f$?
– πr8
Jul 17 '20 at 12:34
• The smoothness condition is the interchangeability of integration and differentiation, which seems to be satisfied for the example I gave, isn't it? except the boundaries, is that the issue? Jul 17 '20 at 12:39
• yes, the boundaries are the issue
– πr8
Jul 17 '20 at 12:59
• The uniform distribution has a support depending on $\theta$, thus a regularity condition is violated. That is to say the CR bound/inequality is not applicable here. Jul 17 '20 at 13:09
• thanyou @πr8 and StubbornAtom. regularity conditions, which I pushed under the carpet came back to bite me. Thanks for the hints. Jul 17 '20 at 13:15