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

Exploiting Dependence over “Metric” Continuous Domains
Recall our special problem:
H0 : µ(xt) = 0 for all t ∈ {1, . . . , T },
HA : µ(xt) = 0 for some t ∈ {1, . . . , T }.
The T test statistics arise as through the discrete approximation of a
signal on an “metric” continuous domain, say [0, S],
0 = x1 < x2 < . . . < xT −1 < xT = S
We thus expect T to become very large as we increase our resolution.
Might be worth “imbedding” the problem in continuous domain
Rather than a collection of T test statistics {Wt : t = 1, . . . , T },
consider an uncountable collection indexed by the interval [0, S].
Take advantage of metric structure and continuity of domain (could
they allow more structured dependence to be further exploited?)
View collection {W (x) : x ∈ [0, S]} as random function (stoch. process)
Victor M. Panaretos (EPFL) Progress on Statistical Issues in Searches SLAC – June 2012 17 / 25