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

Question

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?

Answer 1

The sandwich estimator does not pertain only to maximum likelihood. It is often used with maximum likelihood and if you estimate a correctly specified model then you obtain the information matrix as the variance estimate. However, the sandwich estimator can also be used in the case of optimisation problems, which can be seen in the same context as maximum likelihood (in fact ML is a maximisation problem).

To arrive at the sandwich estimate you have to calculate the covariance matrix and sandwich it with the negative of the Hessian. In the pure context of optimisation you often use the outer product of gradients (OPG) as a way to calculate the Hessian or the information matrix. In your question's abstract form it's a little hard to gather what the authors of the paper were trying to do in practise, so I would not know how the exact calculations ensue at this point, but these are the steps you should take to arrive at the sandwich estimator.