Ok, so to recap, we have a large model (such as ‘we know the marginal sampling probabilities’) and a small model (such as the subset of the large model with \(\mathrm{logit}\,P[Y=1]=x\beta\)). Under the large model, we would use the estimator \(\hat\beta_{L}\), but under the small model there is a more efficient estimator \(\hat\beta_S\). That is, under the small model
\[\sqrt{n}(\hat\beta_S-\beta_0)\stackrel{d}{\to}N(0,\sigma^2)\]
and
\[\sqrt{n}(\hat\beta_L-\beta_0)\stackrel{d}{\to}N(0,\sigma^2+\omega^2)\]
We’re worried that the small model might be slightly misspecified. One test of model misspecification is based on \(D=\hat\beta_S-\hat\beta_L\). Under the small model, \(\sqrt{n}D\stackrel{d}{\to}N(0,\tau^2)\) for some \(\tau^2\). This test isn’t a straw man – for example, DuMouchel and Duncan recommended it in the context of survey regression in a 1983 JASA paper.
If we assume that \(\hat\beta_S\) is (locally, semiparametric) efficient in the small model then \(\tau=\omega\). Now suppose the small model is slightly untrue so that \(\sqrt{n}D\stackrel{d}{\to}N(\Delta,\omega^2)\) with \(\Delta>0\). If, say, \(\Delta=\omega\), then approximately
\[\hat\beta_S\sim N(\omega, \sigma^2)\]
and
\[\hat\beta_L\sim N(0, \sigma^2+\omega^2)\]
so the two estimators have the same asymptotic mean squared error. Since \(\hat\beta_L\) is asymptotically unbiased it would probably be preferred, but the test based on \(D\) has noncentrality parameter 1 and very poor power. If we relied on the test, we would probably end up choosing \(\hat \beta_S\)
So the test based on \(D\) is not very useful if we want to protect against small amounts of model misspecification. We should use a better test.
But sometimes the test based on \(D\) is the most powerful test or not far from it. Since we know what \(\hat\beta_S\) and \(\hat\beta_L\) look like as functionals of the distribution, we could try to maliciously arrange for the model misspecification to be in the direction that maximised \(\hat\beta_S-\hat\beta_L\), and \(D\) would then be the Neyman-Pearson most powerful test – that’s what UMP tests look like for Gaussian shift alternatives. We can’t quite do that, but in large enough sample sizes we can come as close as we need.