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Control of generalized error rates in multiple testing. (English) Zbl 1127.62063
Summary: Consider the problem of testing $$s$$ hypotheses simultaneously. The usual approach restricts attention to procedures that control the probability of even one false rejection, the familywise error rate (FWER). If $$s$$ is large, one might be willing to tolerate more than one false rejection, thereby increasing the ability of the procedure to correctly reject false null hypotheses. One possibility is to replace control of the FWER by control of the probability of $$k$$ or more false rejections, which is called the $$k$$-FWER. We derive both single-step and step-down procedures that control the $$k$$-FWER in finite samples or asymptotically, depending on the situation. We also consider the false discovery proportion (FDP) defined as the number of false rejections divided by the total number of rejections (and defined to be 0 if there are no rejections).
The false discovery rate proposed by Y. Benjamini and Y. Hochberg [J. R. Stat. Soc., Ser. B 57, No. 1, 289–300 (1995; Zbl 0809.62014)] controls $$E$$(FDP). Here, the goal is to construct methods which satisfy, for a given $$\gamma$$ and $$\alpha$$, $$P\{\text{FDP}>\gamma\}\leq\alpha$$, at least asymptotically. In contrast to the proposals of E. L. Lehmann and J. P. Romano [Ann. Stat. 33, No. 3, 1138–1154 (2005; Zbl 1072.62060)], we construct methods that implicitly take into account the dependence structure of the individual test statistics in order to further increase the ability to detect false null hypotheses. This feature is also shared by related work of M. J. van der Laan, S. Dudoit and K. S. Pollard [Stat. Appl. Genet. Mol. Biol. 3, No. 1, Article 15, electronic only (2004; Zbl 1166.62379)], but our methodology is quite different. Like the work of K. S. Pollard and M. J. van der Laan [Int. Multi-Conf. Computer Sci. Eng., METMBS’03 Conference, 3–9 (2003)] and S. Dudoit, M. J. van der Laan and K. S. Pollard [Stat. Appl. Genet. Mol. Biol. 3, No. 1, Article 13, electronic only (2004; Zbl 1166.62338)], we employ resampling methods to achieve our goals. Some simulations compare finite sample performance to currently available methods.

##### MSC:
 62J15 Paired and multiple comparisons; multiple testing 62G09 Nonparametric statistical resampling methods 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference
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