Review Articles

Spatial-sign-based high-dimensional white noises test

Ping Zhao ,

School of Statistics and Data Science, KLMDASR, LEBPS, and LPMC, Nankai University, Nankai District, Tianjin, People's Republic of China

Dachuan Chen ,

School of Statistics and Data Science, KLMDASR, LEBPS, and LPMC, Nankai University, Nankai District, Tianjin, People's Republic of China

Zhaojun Wang

School of Statistics and Data Science, KLMDASR, LEBPS, and LPMC, Nankai University, Nankai District, Tianjin, People's Republic of China

Pages | Received 18 Sep. 2023, Accepted 30 May. 2024, Published online: 07 Jun. 2024,
  • Abstract
  • Full Article
  • References
  • Citations

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. Bernoulli22(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.


  • Cai, J., & Kwan, M. P. (2022). Detecting spatial flow outliers in the presence of spatial autocorrelation. Computers, Environment and Urban Systems96, 101833.
  • Chang, J., Yao, Q., & Zhou, W. (2017). Testing for high-dimensional white noise using maximum cross-correlations. Biometrika104(1), 111–127.
  • Chen, D., Song, F., & Feng, L. (2022). Rank based tests for high dimensional white noise. arXiv:2204.08402.
  • Feng, L., & Liu, B. (2017). High-dimensional rank tests for sphericity. Journal of Multivariate Analysis155, 217–233.
  • Feng, L., Liu, B., & Ma, Y. (2021). An inverse norm sign test of location parameter for high-dimensional data. Journal of Business & Economic Statistics39(3), 807–815.
  • Feng, L., Liu, B., & Ma, Y. (2022). Testing for high-dimensional white noise. arXiv:2211.02964.
  • Feng, L., & Sun, F. (2016). Spatial-sign based high-dimensional location test. Electronic Journal of Statistics10(2), 2420–2434.
  • Feng, L., Zou, C., & Wang, Z. (2016). Multivariate-sign-based high-dimensional tests for the two-sample location problem. Journal of the American Statistical Association111(514), 721–735.
  • Hall, P., & Heyde, C. C. (1980). Martingale limit theory and its application. Academic Press.
  • Hosking, J. R. (1980). The multivariate portmanteau statistic. Journal of the American Statistical Association75(371), 602–608.
  • Huang, X., Liu, B., Zhou, Q., & Feng, L. (2023). A high-dimensional inverse norm sign test for two-sample location problems. Canadian Journal of Statistics51(4), 1004–1033.
  • Li, W. K. (2003). Diagnostic checks in time series. Chapman and Hall/CRC.
  • Li, Z., Lam, C., Yao, J., & Yao, Q. (2019). On testing for high-dimensional white noise. The Annals of Statistics47(6), 3382–3412.
  • Li, W. K., & McLeod, A. I. (1981). Distribution of the residual autocorrelations in multivariate ARMA time series models. Journal of the Royal Statistical Society Series B: Statistical Methodology43(2), 231–239.
  • Lintner, J. (1975). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. In Stochastic optimization models in finance (pp. 131–155). Elsevier.
  • Liu, B., Feng, L., & Ma, Y. (2023). High-dimensional alpha test of linear factor pricing models with heavy-tailed distributions. Statistica Sinica33, 1389–1410.
  • Lütkepohl, H. (2005). New introduction to multiple time series analysis. Springer.
  • Oja, H. (2010). Multivariate nonparametric methods with R: An approach based on spatial signs and ranks. Springer Science & Business Media.
  • Paindaveine, D., & Verdebout, T. (2016). On high-dimensional sign tests. Bernoulli22(3), 1745–1769.
  • Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance19(3), 425–442.
  • Shekhar, S., C. T. Lu, & Zhang, P. (2003). A unified approach to detecting spatial outliers. GeoInformatica7(2), 139–166.
  • Tsay, R. S. (2020). Testing serial correlations in high-dimensional time series via extreme value theory. Journal of Econometrics216(1), 106–117.
  • Wang, L., Peng, B., & Li, R. (2015). A high-dimensional nonparametric multivariate test for mean vector. Journal of the American Statistical Association110(512), 1658–1669.
  • Zhao, P. (2023). Robust high-dimensional alpha test for conditional time-varying factor models. Statistics57(2), 444–457.
  • Zhao, P., Chen, D., & Zi, X. (2022). High-dimensional non-parametric tests for linear asset pricing models. Stat11(1), e490.
  • Zou, C., Peng, L., Feng, L., & Wang, Z. (2014). Multivariate sign-based high-dimensional tests for sphericity. Biometrika101(1), 229–236.

To cite this article: Ping Zhao, Dachuan Chen & Zhaojun Wang (07 Jun 2024): Spatialsign-based high-dimensional white noises test, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2024.2363715

To link to this article: