Looking Elsewhere and Looking Everywhere (a.k.a. Multiple Testing ...
Looking Elsewhere and Looking Everywhere (a.k.a. Multiple Testing ...
Looking Elsewhere and Looking Everywhere (a.k.a. Multiple Testing ...
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Tests Based on Biased Estimators<br />
Let µ <strong>and</strong> K be “smooth” <strong>and</strong> let ˆ λ be a data-tuned balancer of bias &<br />
variance with (ˆλ − λ ∗ )/λ ∗ = OP(T −1/10 ) (λ ∗ the true one). Let ˆQα be<br />
σ K 2 (x)dx<br />
<br />
T ˆλ<br />
⎧<br />
⎛<br />
⎪⎨<br />
−2 log ˆλ + log ⎝ 1<br />
˙K<br />
2π<br />
2 ⎞<br />
(y)dy<br />
⎠ − log<br />
K 2 (y)dy<br />
<br />
−2 log ˆλ<br />
⎪⎩<br />
− log(1−α)<br />
2<br />
⎫<br />
<br />
Then, under {H0 : µ(x) = 0 ∀x ∈ [0, 1]}, <strong>and</strong> ˆbˆ λ (x) = ˆλ 2 1<br />
2 ˆµ′′ (x) K 2dy, P<br />
<br />
<br />
<br />
sup ˆµˆ λ (x) − bˆ λ(x) (x)<br />
x∈[0,1]<br />
<br />
<br />
<br />
> ˆQ1−α<br />
T →∞<br />
−→ α<br />
⎪⎬<br />
.<br />
ˆbˆ λ (x) is a point-wise asymptotic bias correction.<br />
Bonferroni would be conservative, even though failing to account for<br />
bias! (still, this result is asymptotic in T (# of observations))<br />
Victor M. Panaretos (EPFL) Progress on Statistical Issues in Searches SLAC – June 2012 23 / 25<br />
⎪⎭
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