Tests of independence and randomness for arbitrary data using copula-based covariances

TL;DR

This paper introduces copula-based covariance tests for assessing independence and randomness in arbitrary data, providing asymptotic distributions and evaluating finite sample performance through numerical experiments.

Contribution

It develops new copula-based covariance tests for independence and randomness applicable to arbitrary distributions, including their asymptotic properties and optimality under certain alternatives.

Findings

Asymptotic distributions of the tests are derived under null and alternative hypotheses.

Numerical experiments demonstrate the finite sample effectiveness of the proposed tests.

The tests are applicable to both i.i.d. data and time series, regardless of marginal distributions.

Abstract

In this article, we study tests of independence for data with arbitrary distributions in the non-serial case, i.e., for independent and identically distributed random vectors, as well as in the serial case, i.e., for time series. These tests are derived from copula-based covariances and their multivariate extensions using M\"obius transforms. We find the asymptotic distributions of these statistics under the null hypothesis of independence or randomness, as well as under contiguous alternatives. This enables us to find out locally most powerful test statistics for some alternatives, whatever the margins. Numerical experiments are performed for Wald's type combinations of these statistics to assess the finite sample performance.