# How to implement the sandwich estimator in a semi-parametric situation?

I am trying to implement a sandwich estimator described in Zhang et al. (2012, p. 1012) in very brief terms. The information they give is not enough for me to understand what has been actually done, which is why I re-produce the problem here in simplified notation hoping somebody can give guidance on what to do.

We have some population parameter $Q$ which is a many-to-one function of parameter vector $\eta$. In one simple version, $Q$ is an inverse probability weighted (semi-parametric estimator of a) mean:

$$Q(\eta) = n^{-1} \sum_n C_\eta Y / \pi_{C,\eta}$$

where $n$ sample size, $C_\eta$ an indicator function $(0,1)$ depending on $\eta$ which indicates whether $Y$ is taken into the estimator and $\pi_{C,\eta}$ the probability of $C_{\eta}=1$. $Q$ is maximized over $\eta$ using numerical optimization.

For large $n$ the authors argue:

$$n^{1/2} (\hat{Q}(\hat{\eta})-Q(\eta))=n^{1/2} (\hat{Q}(\eta)-Q(\eta)) + o(1)$$

"so that the asymptotic variance of the left-hand side can be approximated by that of the leading term on the right, which can be esimated by the usual sandwich technique (Stefanski & Boos, 2002)." The authors do not give any other information.

As far as I know the sandwich estimator is a likelihood based measure. However, here I do not see a likelihood, since optimization of $Q$ gives an estimate of maximized $Q$, namely $\hat{Q}$, as well as estimates $\hat{\eta}$.

Which steps would I probably need to take to arrive at a sandwich estimator?