Abstract

Testing multi-dimensional white noise has been an important subject of statis

tical inference in time series. Such test in the high-dimensional case becomes an open

problem waiting to be further investigated, especially when the dimension of a time series

is comparable to or even greater than the sample size. To detect an arbitrary form of departure from high-dimensional white noise, a few tests have been developed. Some of these

tests are based on max-type statistics, while others are based on sum-type ones. Despite

the progress, an urgent issue awaits to be resolved: none of these tests is robust to the

sparsity of the serial correlation structure. Motivated by this, we propose a Fisher’s combination test by combining the max-type and the sum-type statistics, taking advantage of

the established asymptotic independence between them. This combination test can achieve

robustness to the sparsity of the serial correlation structure, and combine the advantages of

the two types of tests. We thoroughly study the theoretical properties of the proposed combination test, and demonstrate its advantages over some existing tests through extensive

numerical results and an empirical analysis.

Information

Preprint No.SS-2023-0300
Manuscript IDSS-2023-0300
Complete AuthorsLong Feng, Binghui Liu, Yanyuan Ma
Corresponding AuthorsBinghui Liu
Emailsliubh100@nenu.edu.cn

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Acknowledgments

Feng’s research was partially supported by Shenzhen Wukong Investment Company, Tianjin Science Fund for Outstanding Young Scholar (23JCJQJC00150),

the Fundamental Research Funds for the Central Universities under Grant No.

ZB22000105 and 63233075, the China National Key R&D Program (Grant Nos.

and the National Natural Science Foundation of China Grants (Nos. 12271271,

11925106, 12231011, 11931001 and 11971247). Liu’s research was partially

supported by National Natural Science Foundation of China grant 12171079 and

the China National Key R&D Program grant 2020YFA0714100. Ma’s research

was partially supported by grants from national sciences foundation and national

institute of health.

Supplementary Materials

The Supplementary Material presents the technical details of Remark 2, some

additional simulation results and the technical proofs.

Supplementary materials are available for download.