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As I understand it, the use of Cramer-Rao inequality requires a few regularity conditions, including the parametric family being dominated and swapping of differentiation and integration in computation of expectation of the estimator. How do I prove the latter condition for unbiased estimators?

Mathematically, suppose $\hat{\theta}\left(X\right)$ is an unbiased estimator of $\theta$. I want the proof of the following statement $$\frac{\partial}{\partial\theta}\int \hat{\theta}\left(X\right)p(x,\theta)dx = \int \frac{\partial}{\partial\theta} \left(\hat{\theta}\left(X\right)p(x,\theta)\right)dx$$

My progress : Obviously, $\hat{\theta}\left(X\right)$ being unbiased for $\theta$, the left hand side is $1$. I'm having trouble showing that the right hand side is also equal to $1$. I have tried to right $p(x,\theta)$ in terms of the log-likelihood $L(x,\theta)$ as $e^{L(x,\theta)}$ without success. My calculations reduce to showing that $E_{\theta}\left(\hat{\theta}\left(X\right)\frac{\partial}{\partial\theta}L(X,\theta)\right)=1$.

Any help would be much appreciated. Thank you.

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I'm not sure the statement:

$$\frac{\partial}{\partial\theta}\int \hat{\theta}\left(X\right)p(x,\theta)dx = \int \frac{\partial}{\partial\theta} \left(\hat{\theta}\left(X\right)p(x,\theta)\right)dx$$

Is true in general. For example, letting $P[x|\theta] \sim \textrm{Exp}(\lambda)$, then $\hat{\lambda}(X) = \lambda$ is an unbiased oracle estimator.

We then have that"

\begin{equation} \begin{split} & \frac{\partial}{\partial\lambda}\int \hat{\lambda}\left(X\right)p(x,\lambda)dx = \int \frac{\partial}{\partial\lambda}\left(\hat{\lambda}\left(X\right)p(x,\lambda)\right)dx\\ & \Leftrightarrow 1 = \int\frac{d}{d\lambda}\lambda^2\textrm{Exp}(\lambda x) = \int 2\lambda\textrm{Exp}(\lambda x) + \lambda^2x\textrm{Exp}(\lambda x) dx) = 2 + \lambda^2 \end{split} \end{equation}

The wikipedia page doesn't mention that unbiasedness is a requirement, but does incorporate how they use it in one of the formulations here. The following section mentions exchanging differentiation and integration, but mentions two sufficient methods for showing it can be done.

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