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.

References

• Cai, J., & Kwan, M. P. (2022). Detecting spatial flow outliers in the presence of spatial autocorrelation. Computers, Environment and Urban Systems, 96, 101833. https://doi.org/10.1016/j.compenvurbsys.2022.101833
• Chang, J., Yao, Q., & Zhou, W. (2017). Testing for high-dimensional white noise using maximum cross-correlations. Biometrika, 104(1), 111–127. https://doi.org/10.1093/biomet/asw066
• 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 Analysis, 155, 217–233. https://doi.org/10.1016/j.jmva.2017.01.003
• Feng, L., Liu, B., & Ma, Y. (2021). An inverse norm sign test of location parameter for high-dimensional data. Journal of Business & Economic Statistics, 39(3), 807–815. https://doi.org/10.1080/07350015.2020.1736084
• 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 Statistics, 10(2), 2420–2434. https://doi.org/10.1214/16-EJS1176
• Feng, L., Zou, C., & Wang, Z. (2016). Multivariate-sign-based high-dimensional tests for the two-sample location problem. Journal of the American Statistical Association, 111(514), 721–735. https://doi.org/10.1080/01621459.2015.1035380
• 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 Association, 75(371), 602–608. https://doi.org/10.1080/01621459.1980.10477520
• Huang, X., Liu, B., Zhou, Q., & Feng, L. (2023). A high-dimensional inverse norm sign test for two-sample location problems. Canadian Journal of Statistics, 51(4), 1004–1033. https://doi.org/10.1002/cjs.v51.4
• 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 Statistics, 47(6), 3382–3412. https://doi.org/10.1214/18-AOS1782
• 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 Methodology, 43(2), 231–239. https://doi.org/10.1111/j.2517-6161.1981.tb01175.x
• 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 Sinica, 33, 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. Bernoulli, 22(3), 1745–1769. https://doi.org/10.3150/15-BEJ710
• Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425–442.
• Shekhar, S., C. T. Lu, & Zhang, P. (2003). A unified approach to detecting spatial outliers. GeoInformatica, 7(2), 139–166. https://doi.org/10.1023/A:1023455925009
• Tsay, R. S. (2020). Testing serial correlations in high-dimensional time series via extreme value theory. Journal of Econometrics, 216(1), 106–117. https://doi.org/10.1016/j.jeconom.2020.01.008
• Wang, L., Peng, B., & Li, R. (2015). A high-dimensional nonparametric multivariate test for mean vector. Journal of the American Statistical Association, 110(512), 1658–1669. https://doi.org/10.1080/01621459.2014.988215
• Zhao, P. (2023). Robust high-dimensional alpha test for conditional time-varying factor models. Statistics, 57(2), 444–457. https://doi.org/10.1080/02331888.2023.2180003
• Zhao, P., Chen, D., & Zi, X. (2022). High-dimensional non-parametric tests for linear asset pricing models. Stat, 11(1), e490. https://doi.org/10.1002/sta4.v11.1
• Zou, C., Peng, L., Feng, L., & Wang, Z. (2014). Multivariate sign-based high-dimensional tests for sphericity. Biometrika, 101(1), 229–236. https://doi.org/10.1093/biomet/ast040

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: https://doi.org/10.1080/24754269.2024.2363715