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SUMMARY:Optimal rates for independence testing via U-statistic permutation
  tests - Tom Berrett\, University of Warwick
DTSTART:20210219T160000Z
DTEND:20210219T170000Z
UID:TALK155908@talks.cam.ac.uk
CONTACT:Dr Sergio Bacallado
DESCRIPTION:Independence testing is one of the most well-studied problems 
 in statistics\, and the use of procedures 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 i
 n popularity due to their simplicity and exact Type I error control. In th
 is talk I will present new\, finite-sample results on the power of a new c
 lass of permutation tests\, which show that their power is optimal in many
  interesting settings\, including those with discrete\, continuous\, and f
 unctional data. A simulation study shows that our test for discrete data c
 an significantly outperform the chi-squared for natural data-generating di
 stributions.\n\n \n\nDefining a natural measure of dependence $D(f)$ to be
  the squared $L^2$-distance between a joint density $f$ and the product of
  its marginals\, we first show that there is generally no valid test of in
 dependence that is uniformly consistent against alternatives of the form $
 \\{f: D(f) \\geq \\rho^2 \\}$. Motivated by this observation\, we restrict
  attention to alternatives that satisfy additional Sobolev-type smoothness
  constraints\, and consider as a test statistic a U-statistic estimator of
  $D(f)$. Using novel techniques for studying the behaviour of U-statistics
  calculated on permuted data sets\, we prove that our tests can be minimax
  optimal. Finally\, based on new normal approximations in the Wasserstein 
 distance for such permuted statistics\, we also provide an approximation t
 o the power function of our permutation test in a canonical example\, whic
 h offers several additional insights.\n\n \n\nThis is joint work with Ioan
 nis Kontoyiannis and Richard Samworth.
LOCATION: https://maths-cam-ac-uk.zoom.us/j/92821218455?pwd=aHFOZWw5bzVReU
 NYR2d5OWc1Tk15Zz09
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