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CATEGORIES:Statistics
SUMMARY:Optimal rates for independence testing via U-stati
stic permutation tests - Tom Berrett\, University
of Warwick
DTSTART;TZID=Europe/London:20210219T160000
DTEND;TZID=Europe/London:20210219T170000
UID:TALK155908AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/155908
DESCRIPTION:Independence testing is one of the most well-studi
ed problems in statistics\, and the use of procedu
res such as the chi-squared test is ubiquitous in
the sciences. While tests have traditionally been
calibrated through asymptotic theory\, permutation
tests are experiencing a growth in popularity due
to their simplicity and exact Type I error contro
l. In this talk I will present new\, finite-sample
results on the power of a new class of permutatio
n tests\, which show that their power is optimal i
n many interesting settings\, including those with
discrete\, continuous\, and functional data. A si
mulation study shows that our test for discrete da
ta can significantly outperform the chi-squared fo
r natural data-generating distributions.\n\n \n\nD
efining a natural measure of dependence $D(f)$ to
be the squared $L^2$-distance between a joint dens
ity $f$ and the product of its marginals\, we firs
t show that there is generally no valid test of in
dependence that is uniformly consistent against al
ternatives of the form $\\{f: D(f) \\geq \\rho^2 \
\}$. Motivated by this observation\, we restrict a
ttention to alternatives that satisfy additional S
obolev-type smoothness constraints\, and consider
as a test statistic a U-statistic estimator of $D(
f)$. Using novel techniques for studying the behav
iour of U-statistics calculated on permuted data s
ets\, we prove that our tests can be minimax optim
al. Finally\, based on new normal approximations i
n the Wasserstein distance for such permuted stati
stics\, we also provide an approximation to the po
wer function of our permutation test in a canonica
l example\, which offers several additional insigh
ts.\n\n \n\nThis is joint work with Ioannis Kontoy
iannis and Richard Samworth.
LOCATION: https://maths-cam-ac-uk.zoom.us/j/92821218455?pwd
=aHFOZWw5bzVReUNYR2d5OWc1Tk15Zz09
CONTACT:Dr Sergio Bacallado
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