The length heuristic for simultaneous hypothesis tests

We observe a sequence of test statistics 9~ = (Tlt T2,..., 7}), each of which is approximately N(0,1) under the null hypothesis Ho, but which are correlated with each other. Not being certain which 7} is best, we use the test statistic 7 ^ = maxfTj, T2,...,Tj}. For a given observed value of 7 ^ , say c, what is the significance probability pr(7^iax > c)? Define the length of 9~ to be E ' ^ a r c c o s ^ j j ) , where Pj-ij is the null hypothesis correlation between 7}_j and 7}. Hotelling's theorem on the volume of tubes leads to a length-based bound on pr(7Jnax > 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(7^nmx > c).