Asymptotic Normality for GEE Parameters

Question

In the famous Liang and Zeger 1986 paper on GEEs https://www.jstor.org/stable/2336267?seq=9, they sketch a proof using the standard m-estimator arguments: (unstated) regularity conditions + first-order Taylor expansion + somehow a vanishing remainder in probability (covered by regularity conditions) + routine convergence arguments to prove asymptotic normality at rate $\sqrt{n_c}$, where $n_c$ is the number of clusters. What version of the Central Limit Theorem (CLT) for independent data is typically cited to enable asymptotic normality here? I feel like I never see this pointed out specifically anywhere.

To elaborate a bit further, let $\varphi_{n_i}(Z_i;\theta)$ where the $Z_1,...,Z_{n_c}$ are mutually independent group level data such that $E(\varphi_{n_i}(Z_i;\theta))=0$ for all $i=1,...n_c$ and $n_i$ is cluster size. Suppose solving the unbiased estimating equations

$$\sum_{i=1}^{n_c} \varphi_{n_i}(Z_i;\theta) = 0$$

gives you the GEE estimators. For simplicity sake, assume the variance of the random effects is known. The $\varphi_{n_i}(Z_i;\theta)$ are independent and of the same dimension BUT they are only mutually independent; indeed given $n_i$ is not constant, they will in general not be identically distributed. The proof of asymptotic normality of the coefficients requires (among other things) that $\sqrt{n_c}\left (\frac{1}{n_c} \sum_{i=1}^{n_c} \varphi_{n_i}(Z_i;\theta)\right )$ converges in distribution to a mean zero multivariate normal. What version of CLT is typically appealed to to make this statement?

Answer 1

The central limit theorem is just the Lyapunov or Lindeberg CLT, together with the result that this applies to vectors (of fixed finite dimension).

The application to vectors comes from the Cramér-Wold theorem that a sequence of vectors $Z_n$ converges to a Normal vector $Z$ if and only if $b'Z_n\stackrel{d}{\to} b'Z$ for every unit vector $b$.

We can now consider the univariate case of the CLT, we just have to check that whatever tail conditions we impose on the univariate linear combination are implied by reasonable criteria on the vectors.

Since the Lyapunov CLT conditions are in terms of moments, they translate conveniently: the univariate sequence $b'Z_n$ have bounded third moments for every $b$ if and only if each element of the vector $Z_n$ has bounded third moments. For your $\varphi_{n_i}$ that's going to follow from a moment condition on the individual-observation contributions to $\varphi$ and a uniform bound on the sizes $n_i$ of clusters.

(If you want to know how the correlation parameters can be estimated and why it doesn't matter if they are wrong (in some sense), the argument for that is written out here