Bootstrap and permutation tests of independence for point processes

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

Testing independence for multivariate time series via the auto-distance correlation matrix

We introduce the matrix multivariate auto-distance covariance and correlation functions for time series, discuss their interpretation and develop consistent estimators for practical implementation. We also develop a test of the independent and identically distributed hypothesis for multivariate time series data and show that it performs better than the multivariate Ljung–Box test. We discuss computational aspects and present a data example to illustrate the method

An Updated Literature Review of Distance Correlation and Its Applications to Time Series

The concept of distance covariance/correlation was introduced recently to characterise dependence among vectors of random variables. We review some statistical aspects of distance covariance/correlation function, and we demonstrate its applicability to time series analysis. We will see that the auto‐distance covariance/correlation function is able to identify non‐linear relationships and can be employed for testing the i.i.d. hypothesis. Comparisons with other measures of dependence are included

Fourier methods for testing multivariate independence

Recently a power study of some popular tests for bivariate independence based on ranks has been conducted. An alternative class of tests appropriate for testing not only bivariate, but also multivariate independence is developed, and their small-sample performance is studied. The test statistics employ the familiar equation between the joint characteristic function and the product of component characteristic functions, and may be written in a closed form convenient for computer implementation. Simulations on a distribution-free version of the new test statistic show that the proposed method compares well to standard methods of testing independence via the empirical distribution function. The methods are applied to multivariate observations incorporating data from several major stock-market indices. Issues pertaining to the theoretical properties of the new test are also addressed. © 2007 Elsevier B.V. All rights reserved