Spatial-sign-based high-dimensional white noises test

In this study, we explore the problem of hypothesis testing for white noise in high-dimensional settings, where the dimension of the random vector may exceed the sample sizes. We introduce a test procedure based on spatial-sign for high-dimensional white noise testing. This new spatial-sign-based test statistic is designed to emulate the test statistic proposed by Paindaveine and Verdebout [(2016). On high-dimensional sign tests. Bernoulli, 22(3), 1745-1769.], but under a more generalized scatter matrix assumption. We establish the asymptotic null distribution and provide the asymptotic relative efficiency of our test in comparison with the test proposed by Feng et al. [(2022). Testing for high-dimensional white noise. arXiv:2211.02964.] under certain specific alternative hypotheses. Simulation studies further validate the efficiency and robustness of our test, particularly for heavy-tailed distributions.