If $X=(X_1,X_2,...,X_n)$ follows i.i.d. $f_\theta()$, and we have to estimate $g(\theta)$, then
$CRLB=\frac{{(g'(\theta))}^2}{nE_\theta\ {[\frac{\partial lnf_\theta({X_i})}{\partial \theta}\ ]}^2}$
$= \frac{{(g'(\theta))}^2}{E_\theta\ {[\frac{\partial lnp_\theta({X})}{\partial \theta}\ ]}^2}$
where $p_\theta(x)$ is the joint distribution of $(X_1,X_2,...,X_n)$.
But when $X=(X_1,X_2,...,X_n)$ follows i.i.d. $U(0,\theta)$ distribution, and $g(\theta)=\theta$, the first form gives $CRLB=\frac{{\theta}^2}{n}$, while the second form gives $CRLB= \frac{{\theta}^2}{n^2}$.
Any help is appreciated. Thanks in advance.
