Bootstrap and permutation tests of independence for point processes

TL;DR

This paper develops non-parametric bootstrap and permutation tests for independence in point processes, particularly applied to spike train analysis in neuroscience, with proven consistency and good performance in simulations.

Contribution

It introduces new rescaled U-statistics-based tests for independence in point processes, with theoretical guarantees and practical comparison to existing methods.

Findings

Tests are asymptotically correct in size.

Tests are consistent against all alternatives.

Simulation shows competitive performance.

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 non-parametric 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. to Wasserstein's metric, which induce 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.