Let $X = (X_1,\cdots, X_n)$ where $X_1,\cdots,X_n$ be i.i.d from the uniform distribution $U(0,\theta)$ with $\theta>0$. I was asked to show the regularity condition of the Cramer-Rao lower bound:
$$\frac{\partial}{\partial\theta}\int h(x)f_\theta(x)dv = \int h(x)\frac{\partial}{\partial\theta} f_\theta(x)dv$$
does not hold when $h(x) = X_{(n)}$. Here is my attempt:
$f_\theta(x) = \prod_{i=1}^n\frac{1}{\theta}I_{(0,\theta)}(x) = \frac{1}{\theta^n}I_{(0,\theta)}(x_{(n)})$, then $\frac{\partial}{\partial\theta}f_{\theta}(x) =-\frac{n}{\theta^{n+1}}, 0 < x_{(n)} < \theta$. Therefore,
$$\int_0^\theta -x_{(n)}\frac{n}{\theta^{n+1}}dx_{(n)} = -\frac{n}{2\theta^{n-1}}$$ and $$\frac{\partial}{\partial\theta}\int_{0}^\theta x_{(n)}\frac{1}{\theta_n}dx_{(n)} = \frac{-n+2}{2\theta^{n-1}}$$
for which both of them are different. Am I doing this correctly ? I am a bit skeptical if I can integrate w.r.t to $x_{(n)}$.Let me know if I did anything wrongly.
