Looking Elsewhere and Looking Everywhere (a.k.a. Multiple Testing ...

Exploiting Dependence Between Test Statistics
Westfall-Young Single Step Permutation Procedure
Under the complete H0, the joint distribution of (Y1, . . . , YT ) is the
same as the distribution of g(Y1, . . . , YT ), for any permutation g,
g(Y1, . . . , YT ) = (Y π(1), . . . , Y π(T )),
where π(·) : {1, . . . , T } → {1, . . . , T } is a bijection.
Therefore, under H0, the distribution of W ∗ = sup t Wt can be
approximated, by considering all possible permutations of
(Y1, . . . , YT ) (re-calculating the supremum of individual test statistics
for each permutation g in the set of all T ! permutations, say G).
This will give us approximate quantiles for the distribution of
W ∗ = sup t Wt under H0, upon which we can base testing
Q1−α ≈ ˆQ1−α := inf
⎧
⎨
1
w ∈ R : 1
⎩ T !
W ∗ g ≤ w ⎫
⎬
≥ 1 − α
⎭
Victor M. Panaretos (EPFL) Progress on Statistical Issues in Searches SLAC – June 2012 15 / 25
g∈G