ENH GMM joint inference for mean and variance in mean models with auxiliary variance

In RLM, nonlinear RNLM, OLS, GLM quasi-mle we estimate the mean model with an auxiliary variance estimator.
Inference only looks at mean parameters using asymptotic orthogonality of mean and variance (inference for mean parameters is asymptotically correct taking variance estimates as non-random.)

There is not necessarily a joint objective function for the joint optimization of mean and variance parameters.
Symptomatic case is in my attempts for robust nonlinear estimator with conflict between M-estimator for mean with auxiliary variance estimation and joint optimization of mean and variance.

The idea here is to get inferential statistics for the joint mean-variance parameter vector by reformulating it as a GMM problem, i.e. stacking both mean and variance moment conditions.

This should allow us to compare models with joint objective functions with models that don't have a joint objective function (and variance is defined either as moment condition or solves an auxiliary optimization problem with separate objective function).

currently just an idea, I have not tried out anything yet.

Also,
If we impose block diagonality in GMM weights/covariance, then mean parameter inference should not be affected.
We only gain inference for variance as extra result.

Possibly related:
Two-stage models like Heckman's selection model, where we could get joint results table for first and second stage parameters from the GMM representation. We would have same structure of params table as for full MLE.
Second stage inference always depends on first stage results (Murphy-Topel).

see also issues:
#7436
...

The issue names RLM, nonlinear RNLM, OLS, GLM quasi-MLE, and GMM inference paths but no files or tests. Start by reviewing those entry points and related issue #7436, then define how stacked mean and auxiliary-variance moments should provide joint inference, including the block-diagonal case.