PhD thesis

PhD thesis:

I defended my PhD thesis on November 16th 2016 at the University of Nice Sophia Antipolis. It was awarded the Marie-Jeanne Laurent-Duhamel Prize from the French Statistical Society.

  • Title: Tests of independence by bootstrap and permutation: an asymptotic and non-asymptotic study. Application to Neurosciences.
  • Referees: Gilles Blanchard (Potsdam University, Allemagne), Robert E. Kass (Carnegie Mellon University, Pittsburg, États-Unis).
  • Jury: Sylvain Arlot, Patrice Bertail, Gilles Blanchard, Jean-François Coeurjolly, Magalie Fromont, Robert E. KASS, Oleg Lepski, Patricia Reynaud-Bouret.
  • Keywords: Independence test, bootstrap, permutation, randomization, U-statistics, point processes, neuroscience, spike train analysis, synchronization, Unitary Events, trial-shuffling, multiple testing, concentration inequalities, uniform separation rates, adaptive tests, wavelets, weak Besov bodies, aggregated tests.
  • Abstract: Initially motivated by synchrony detection in spike train analysis in neuroscience, the purpose of this thesis is to construct new non-parametric tests of independence adapted to point processes, with both asymptotic and non-asymptotic good performances. On the one hand, we construct such tests based on bootstrap and permutation approaches. Their asymptotic performance are studied in a point process framework through the analysis of the asymptotic behaviors of the conditional distributions of both bootstrapped and permuted test statistics, under the null hypothesis as well as under any alternative. A simulation study is performed verifying the usability of these tests in practice, and comparing them to existing classical methods in neuroscience. We then focus on the permutation tests, well known for their good properties in terms of non-asymptotic level. Their p-values, based on the delayed coincidence count, are implemented in a Benjamini-Hochberg type multiple testing procedure, called Permutation Unitary Events method, to detect the synchronization occurrences between two spike trains in neuroscience. The practical validity of the method is verified on a simulation study before being applied on real data. On the other hand, the non-asymptotic performances of the permutation tests are studied in terms of uniform separation rates. A new aggregated independence testing procedure based on the permutation approach, and a wavelet thresholding method is developed in the density framework. Classically, concentration inequalities are necessary to sharply control the quantiles. Based on Talagrand’s fundamental inequalities for random permutations, we provide a new Bernstein-type concentration inequality for randomly permuted sums. In particular, it allows us to upper bound the uniform separation rate of the aggregated procedure over particular classes of functions, namely weak Besov spaces, with respect to the quadratic metric and deduce that, in view of the literature, this procedure seems to be optimal and adaptive in the minimax sense.
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