Are You All Normal? It Depends!

Embargo End Date
2023-07-07

Type
Article

Authors
Chen, Wanfang
Genton, Marc G.

KAUST Department
Statistics Program
Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
Extreme Computing Research Center

Preprint Posting Date
2020-08-25

Online Publication Date
2022-07-07

Print Publication Date
2023-04

Date
2022-07-07

Abstract
The assumption of normality has underlain much of the development of statistics, including spatial statistics, and many tests have been proposed. In this work, we focus on the multivariate setting and first review the recent advances in multivariate normality tests for i.i.d. data, with emphasis on the skewness and kurtosis approaches. We show through simulation studies that some of these tests cannot be used directly for testing normality of spatial data. We further review briefly the few existing univariate tests under dependence (time or space), and then propose a new multivariate normality test for spatial data by accounting for the spatial dependence. The new test utilises the union-intersection principle to decompose the null hypothesis into intersections of univariate normality hypotheses for projection data, and it rejects the multivariate normality if any individual hypothesis is rejected. The individual hypotheses for univariate normality are conducted using a Jarque–Bera type test statistic that accounts for the spatial dependence in the data. We also show in simulation studies that the new test has a good control of the type I error and a high empirical power, especially for large sample sizes. We further illustrate our test on bivariate wind data over the Arabian Peninsula.

Citation
Chen, W., & Genton, M. G. (2022). Are You All Normal? It Depends! International Statistical Review. Portico. https://doi.org/10.1111/insr.12512

Acknowledgements
This research was supported by the National Key Research and Development Program of China (2021YFA1000101), Zhejiang Provincial Natural Science Foundation of China (LZJWY22E090009), Natural Science Foundation of Shanghai (22ZR1420500), and the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, ECNU and King Abdullah University of Science and Technology (KAUST).

Publisher
Wiley

Journal
International Statistical Review

DOI
10.1111/insr.12512

arXiv
2008.10957

Additional Links
https://onlinelibrary.wiley.com/doi/10.1111/insr.12512

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