ENH question score test for mean, wald or score variance estimate

[josef-pkt]

something to look at, but I don't know any details yet

standard ztest and t-test use the "wald" variance as variance estimate

under score test assumptions we should use the variance estimated under the null, e.g. null mean instead of observed mean to compute variance.

This is what the model score_test currently does because the full model is re-estimated under the null restriction.

There are some issues in the literature about choosing the variance, but I never looked closely at that.

e.g. "exact" test would treat variance as nuisance parameter and would need to use some approach to accommodate that
(I guess sup test will not work because domain of variance is unbound)

The opposite of making var depend on null versus alternative
Part of the literature on LR tests, F-tests recommends using the same variance estimate between null and alternative hypothesis to avoid numerical problems.

Partially related: incorporating distributional assumptions as maintained assumption
test of variance are more accurate if normal distribution is assumed for the 4th moment (if normality can be assumed) than tests that are more robust and estimate 4th moments.

parametric distributional assumptions:
We are using poisson variance in poisson test instead of using estimated variance as in a t-test. The latter, or a robust cov_type, is more appropriate if we cannot rely on the poisson assumption for the variance.

also related
local alternatives:
Under the assumption of local alternatives for deriving the test statistic and it's distribution, the variance estimate is always the same asymptotic value for null and sequence of alternative. Asymptotically it doesn't matter in this case.