Another view of the ‘nearly true’ model

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 $logitP[Y=1]=x\beta$).  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.