I'm here looking for assurance that my interpretation is correct. Let the likelihood function under consideration be a conditional likelihood given by


where $r$ is some random variable, $x$ is a particular realization of another random variable that is given and $\theta$ is a parameter we wish to estimate. To find the Cramer-Rao lower bound (CRLB) for any unbiased estimator $\hat{\theta}$ of $\theta$ we need to compute

$$\text{var}\left(\hat{\theta}\right) \geq \frac{1}{-E_{r|x}\left[\frac{\partial^2 \ln p(r|x;\theta)}{\partial \theta^2} | x\right]}$$

where I've chosen to use a conditional expectation in this step because the original likelihood function was conditioned on $x$. In all the standard texts this expectation is always the standard expectation with respect to the data $r$. Would someone confirm that the conditional expectation is the correction version to use at this step?


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