Rank Based Tests for High Dimensional White Noise

TL;DR

This paper introduces distribution-free, rank-based tests for high-dimensional white noise that are robust to heavy tails, do not require finite moments, and can detect complex autocorrelation structures.

Contribution

It proposes novel rank-based testing methods for high-dimensional white noise, including the first asymptotic distribution results allowing cross-sectional dependence.

Findings

Asymptotic null distributions are established for all test families.

Degenerate U-statistics based test detects nonlinear autocorrelations.

The tests are rate optimal and robust to heavy tails.

Abstract

The development of high-dimensional white noise test is important in both statistical theories and applications, where the dimension of the time series can be comparable to or exceed the length of the time series. This paper proposes several distribution-free tests using the rank based statistics for testing the high-dimensional white noise, which are robust to the heavy tails and do not quire the finite-order moment assumptions for the sample distributions. Three families of rank based tests are analyzed in this paper, including the simple linear rank statistics, non-degenerate U-statistics and degenerate U-statistics. The asymptotic null distributions and rate optimality are established for each family of these tests. Among these tests, the test based on degenerate U-statistics can also detect the non-linear and non-monotone relationships in the autocorrelations. Moreover, this is the first result on the asymptotic distributions of rank correlation statistics which allowing for the cross-sectional dependence in high dimensional data.