# 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 $$\mathrm{logit}\,P[Y=1]=x\beta$$).  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.