# How to derive the variance of a point estimator that contains another estimator?

I am trying to derive an variance estimator $var(\hat{\theta}(\mu,x))$ for an point estimator $\hat{\theta}(\mu,x)$, where $\mu$ is the true mean, $x$ is the data.

Now I need to replace the true mean $\mu$ with the sample mean $\hat{\mu}$. Then what is the general approach to incorporate this extra variability of sample mean $\hat{\mu}$ in computing $var(\hat{\theta}(\mu,x))$?

Any reference, comment or suggestion will be appreciated!

• Use the delta method, based on a 2 variable Taylor expansion. – kimchi lover Aug 22 '17 at 15:17
• Thanks, Kimchi. I thought about that. But this scenario is not a combination of two different estimators. it is one is nested within the other one. – Vincent Aug 22 '17 at 16:57
• As in, $\hat\theta$ explicitly depends on (say) $\bar x$ and $\mu$, and $\hat\mu$ also depends on $\bar x$ and some other $x$-based sample average? Unroll it into an expression $\hat \theta = f( \overline{\text{this}}, \overline{\text{that}})$... – kimchi lover Aug 22 '17 at 17:14