## Annals of Statistics

### Bootstrap and permutation tests of independence for point processes

#### Abstract

Motivated by a neuroscience question about synchrony detection in spike train analysis, we deal with the independence testing problem for point processes. We introduce nonparametric test statistics, which are rescaled general $U$-statistics, whose corresponding critical values are constructed from bootstrap and randomization/permutation approaches, making as few assumptions as possible on the underlying distribution of the point processes. We derive general consistency results for the bootstrap and for the permutation w.r.t. Wasserstein’s metric, which induces weak convergence as well as convergence of second-order moments. The obtained bootstrap or permutation independence tests are thus proved to be asymptotically of the prescribed size, and to be consistent against any reasonable alternative. A simulation study is performed to illustrate the derived theoretical results, and to compare the performance of our new tests with existing ones in the neuroscientific literature.

#### Article information

Source
Ann. Statist., Volume 43, Number 6 (2015), 2537-2564.

Dates
Revised: May 2015
First available in Project Euclid: 7 October 2015

https://projecteuclid.org/euclid.aos/1444222084

Digital Object Identifier
doi:10.1214/15-AOS1351

Mathematical Reviews number (MathSciNet)
MR3405603

Zentralblatt MATH identifier
1327.62454

#### Citation

Albert, Mélisande; Bouret, Yann; Fromont, Magalie; Reynaud-Bouret, Patricia. Bootstrap and permutation tests of independence for point processes. Ann. Statist. 43 (2015), no. 6, 2537--2564. doi:10.1214/15-AOS1351. https://projecteuclid.org/euclid.aos/1444222084

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#### Supplemental materials

• Technical results and proofs of “Bootstrap and permutation tests of independence for point processes”. This Supplement consists of all the proofs. It also contains some additional results about nondegeneracy and the empirical centering assumption.