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Journal Article

# The Length Heuristic for Simultaneous Hypothesis Tests

Biometrika
Vol. 84, No. 1 (Mar., 1997), pp. 143-157
Stable URL: http://www.jstor.org/stable/2337562
Page Count: 15

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## Abstract

We observe a sequence of test statistics mathscrT = (T1, T2,..., TJ), each of which is approximately N(0, 1) under the null hypothesis H0, but which are correlated with each other. Not being certain which Tj is best, we use the test statistic Tmax = max T1, T2,..., TJ. For a given observed value of Tmax, say c, what is the significance probability or $(T_{\max} > c)$?. Define the length of mathscrT to be $\sum^J-j_{=2} \operatorname{arccos} (p_j-J)$, where PJ-J is the null hypothesis correlation between Tj-1 and Tj. Hotelling's theorem on the volume of tubes leads to a length-based bound on pr$(T_{\max} > c)$ that usually beats the Bonferroni bound. It is easy to improve Hotelling's bound. Several examples show that in favourable circumstances the improved bounds can be good approximations to the actual value of pr$(T_{\max} > c)$.

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