If we do two one-sided tests instead of a two-sided test, can we get extra power by using a better multiple testing correction than Bonferroni? A standard problem in statistics would be something like the following.
$$
H_0: \theta = \theta_0
$$
$$
H_a: \theta \ne \theta_0
$$
This is a two-sided test.
What if we did the following two tests?
$$
_1H_0: \theta = \theta_0
$$
$$
_1H_a: \theta > \theta_0
$$
$$
_2H_0: \theta = \theta_0
$$
$$
_2H_a: \theta < \theta_0
$$
The complaint that I see is that this raises $\alpha$, since there are multiple testing issues at play, and then if we consider multiple tests, this becomes equivalent to the two-sided test.
There is more than one way to adjust a p-value, and that approach, to me, implies a simple Bonferroni correction.
There are better adjustments than Bonferroni.
Could we keep $\alpha$ at the level we want (say $0.05$ or $0.01$) but increase power by using a better correction than Bonferroni?
 A: One practical issue is that the two alternative sub-hypotheses in this context are are mutually exclusive and the experimenter is really just interested in rejecting one of them. So for example, suppose the Holm procedure is applied instead of the Bonferroni, then you either reject the test with the lowest p-value if this is less than $\alpha/2$ or reject none at all, which is exactly the same situation if you used Bonferroni since it is not possible to reject both hypotheses anyway.
You can try other adjustments but each would have its own added assumptions and requirements, some of which are not applicable to this setup where there is high negative dependence between the two hypotheses. For example, Sidak's procedure may fail to control family-wise error rate under negative dependence.
A: Yes. Assuming you have more than one such two-sided test, you may consider:
The Holm-Šidak adjustment gives much more powerful sequential rejection control over the family-wise error rate under dependency.
Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(65–70), 1979.
The Benjamini-Hochberg adjustment gives even more powerful control of the false discovery rate.
Benjamini, Y., & Hochberg, Y. (1995). Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing. Journal of the Royal Statistical Society. Series B (Methodological), 57(1), 289–300.
