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 />
Often the test statistics W (t) are based on a biased estimator.<br />
Means that W (t) may not be centred under H0<br />
↩→ example: nonparametric estimation of µ(x), assuming it is smooth,<br />
but without assuming any specific parametric form.<br />
e.g. ssume µ(·) : [0, 1] → R has Lipschitz second derivative, <strong>and</strong> observe<br />
Yt = µ(xt) + εt, t = 1, . . . , T .<br />
with εt assumed iid variance σ 2 . Test H0 : µ(x) = 0 ∀x ∈ [0, 1].<br />
A classical estimator of µ is a kernel estimator (convolution estimator)<br />
ˆµλ(x) = 1<br />
λT<br />
T<br />
<br />
x − xt<br />
YtK<br />
λ<br />
t=1<br />
<br />
, K a centred symmetric pdf on [−1, 1]<br />
Victor M. Panaretos (EPFL) Progress on Statistical Issues in Searches SLAC – June 2012 21 / 25
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