Suppose our model has a nuisance parameter $\eta_0$ of which we possess a consistent estimator $\hat{\eta}_0$.

We obtain an estimator $\hat{\theta}$ of a parameter of interests $\theta$ by finding the $\theta$ that solves the estimating equation

$$S_n(\theta, \hat{\eta}) = 0 $$

However, if we know $\eta_0$, can we obtain a better estimator? Specifically, consider the following.

Question: Under which conditions is $\hat{\theta}$ asymptotically equivalent to $\tilde{\theta}$, where $\tilde{\theta}$ is an estimator obtained by solving the estimating equation

 

$$S_n(\theta, \eta_0) = 0 $$

which requires $\eta_0$ to be known?

Note that conditions for the consistency and $\sqrt{n}$-consistency of $\hat{\theta}$ have been provided in other posts.

Suppose our model has a nuisance parameter $\eta_0$ of which we possess a consistent estimator $\hat{\eta}_0$.

We obtain an estimator $\hat{\theta}$ of a parameter of interests $\theta$ by finding the $\theta$ that solves the estimating equation

$$S_n(\theta, \hat{\eta}) = 0 $$

However, if we know $\eta_0$, can we obtain a better estimator? Specifically, consider the following.

Question: Under which conditions is $\hat{\theta}$ asymptotically equivalent to $\tilde{\theta}$, where $\tilde{\theta}$ is an estimator obtained by solving the estimating equation

$$S_n(\theta, \eta_0) = 0 $$

which requires $\eta_0$ to be known?

Note that conditions for the consistency and $\sqrt{n}$-consistency of $\hat{\theta}$ have been provided in other posts.