On a variation of the Hausman test
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In econometric, Hausman test is frequently used to test if two estimators are both consistent for an unknown model coefficient, under the condition that one estimator is more efficient than the other. In mathematical terms, suppose we have two estimators $\hat\beta_1$ and $\hat\beta_2$ for a model coefficient $\beta$, and $\hat\beta_1$ is asymptotically efficient (achieving the Cramer-Rao lower bound), according to Wikipedia, the testing statistics of the Hausman test is
\[(\hat\beta_2-\hat\beta_1)^\top\mbox{Var}(\hat\beta_2-\hat\beta_1)^{\dagger}(\hat\beta_2-\hat\beta_1)\]where $\dagger$ denotes the Moore-Penrose pseudo inverse. By Lemma 2.1 of Hausman (1978, Econometrica) and the condition that $\hat\beta_1$ is asymptotically efficient, $\hat\beta_2-\hat\beta_1$ and $\hat\beta_1$ are asymptotically uncorrelated, so
\[\mbox{Var}(\hat\beta_2-\hat\beta_1) \to\lim_{n\to\infty}\mbox{Var}(\hat\beta_2)-\lim_{n\to\infty}\mbox{Var}(\hat\beta_1) >0,\]where the “$>0$” holds because $\hat\beta_1$ is more efficient than $\hat\beta_2$. The Hausman test is useful when one suspects that $\hat\beta_1$ is inconsistent (but efficient) while $\hat\beta_2$ is consistent. If one fails to reject the Hausman test, then it may be concluded $\hat\beta_1$ is more favorable because the Hausman test shows both $\hat\beta_1$ and $\hat\beta_2$ are consistent; see this post.
A friend considers a different scenario. Both estimators $\hat\beta_1$ and $\hat\beta_2$ are efficient, and one wants to test whether $\hat\beta_2$ is consistent given that $\hat\beta_1$ is consistent. This amounts to ask whether $\hat\beta_1$ and $\hat\beta_2$ are both consistent and asymptotically efficient. One may question why we should bother with $\hat\beta_2$ given that $\hat\beta_1$ is already a good estimator. The reason may be that $\hat\beta_2$ could be derived from $\hat\beta_1$ using some data normalization that is believed to be efficiency invariant, but $\hat\beta_2$ has a better practical performance, so one may be curious if $\hat\beta_2$ remains consistent.
Can one still apply the Hausman test? The Hausman test is impotent in such scenario because if both $\hat\beta_1$ and $\hat\beta_2$ are asymptotically efficient, so by a similar argument like (2),
\[\mbox{Var}(\hat\beta_1-\hat\beta_2) \to 0.\]Hence, the null distribution of the Hausman test degenerates, and fails to provide a good way for hypothesis testing.
For remedy, one may need to look beyond the difference into ratio statistics
\[R = \frac{\hat\beta_1}{\hat\beta_2},\]and check if $R$ is sufficiently close to 1. There are theoretical results about the limiting distribution of $R$, coined as ratio distribution. Unfortunately, such distribution usually require to estimate many nuisance parameters, such as $\beta$ and the correlation between $\hat\beta_1$ and $\hat\beta_2$. A way to resolve the problem may be to bootstrap $R$ by bootstrapping $\hat\beta_1$ and $\hat\beta_2$, i.e. resampling data without replacement and repeatedly computing the two estimators. Since previous literature has shown that $R$ has a non-degenerate limiting distribution, possibly one could show that the bootstrap samples of $R$ has the same limiting distribution conditional on the data. This conjecture is left for future research.