What are the Mild Regularity Conditions in the context of GEEs?

Question

I was reading the Wikipedia Article on Generalized Estimating Equations, where I came across the following sentence:

Parameter estimates from the GEE are consistent even when the covariance structure is misspecified, under mild regularity conditions.

In this context, what is meant by mild regularity conditions?

Answer 1

Regularity conditions in statistics usually refer to requirements that functions or groups of functions (usually probability density functions) "behave well" in various senses. These are assumptions made in proofs of statements that are thought to hold in most practical cases and often not explicitly mentioned when giving statements of theorems. "Mild" or "weak" in reference to regularity conditions essentially means "we expect these regularity conditions to almost always hold in practice."

For example (roughly taken from Wikipedia), the Cramér-Rao lower bound states that if we have a random variable $X$ with probability density function $f(x;\theta)$, the variance of any unbiased estimator of $\theta$ is bounded below by the reciprocal of the Fisher information $I(\theta)$. However, the article notes that there are two additional conditions: the Fisher information must always be defined, and the operations of integration with respect to $x$ and differentiation with respect to $\theta$ can be interchanged in the expectation of the estimator of $\theta$.

The most common analytical regularity condition I've come across is some variation of exchangeability of limits. For example, we might require that we are able to:

• change the order of integration,
• switch the order of differentiation and integration
• switch the order of a summation or integration and a limit.

Other regularity conditions more specific to statistics include:

• the pdf must be differentiable (or twice, or thrice differentiable),
• the pdfs of a set of random variables must have common support
• the parameter space is open in $\mathbb{R}^k$
• the Fisher Information is always defined.

A richer list can be found here (PDF).

As for the regularity conditions for the consistency of GEE estimates under misspecified covariance structures, it is well-known that the actual model for the mean has to be correct (e.g. if the data is Poisson you must use the log link), but I don't know the more technical conditions. Agresti's Categorical Data Analysis lists a few papers relating to this topic, including

• Firth, D. 1993. Recent developments in quasi-likelihood methods. Proc. ISI 49th Session*, pp. 341-358.
• McCullagh, P. 1983. Quasi-likelihood functions. Ann. Stat. 11:59-67.