, Volume 28, Issue 4, pp 1092–1112 | Cite as

An \({{\varvec{L}}}^{2}\)-norm-based test for equality of several covariance functions: a further study

  • Jia Guo
  • Bu Zhou
  • Jianwei Chen
  • Jin-Ting ZhangEmail author
Original Paper


For the multi-sample equal covariance function (ECF) testing problem, Zhang (Analysis of variance for functional data, CRC Press, Boca Raton, 2013) proposed an \(L^{2}\)-norm-based test. However, its asymptotic power and finite sample performance have not been studied. In this paper, its asymptotic power is investigated under some mild conditions. It is shown that the \(L^2\)-norm-based test is root-n consistent. In addition, intensive simulation studies are conducted to evaluate its finite sample performance against several existing competitors. In particular, they demonstrate that in terms of size control and power, the \(L^{2}\)-norm-based test outperforms the dimension-reduction-based tests proposed by Panaretos et al. (J Am Stat Assoc 105(490):670–682, 2010) and Fremdt et al. (Scand J Stat 40(1):138–152, 2013), respectively, when functional data are less correlated or when the covariance function differences of functional data contain mainly high-frequency signals. Two real data applications are presented to illustrate the \(L^2\)-norm-based test.


\(L^{2}\)-norm-based test Asymptotic power Functional data analysis Multi-sample equal covariance function testing 

Mathematics Subject Classification

62G10 62-07 



Guo was financially supported by the Australian Research Council (ARC) Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). Zhou was financially supported by the First Class Discipline of Zhejiang—A (Zhejiang Gongshang University—Statistics). Zhang was financially supported by the National University of Singapore Academic Research Grant R-155-000-175-114. The first author would like to thank Professor Wen-Lung Shiau, the Advanced Data Analysis Center (PLS-SEM of Zhejiang University of Technology), for his support on this research. The authors thank the Editor-in-Chief and two reviewers for their invaluable comments and suggestions which helped improve the presentation substantially.

Supplementary material

11749_2018_617_MOESM1_ESM.pdf (143 kb)
Supplementary material 1 (pdf 143 KB) (623 kb)
Supplementary material 2 (zip 622 KB)


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2018

Authors and Affiliations

  • Jia Guo
    • 1
  • Bu Zhou
    • 2
  • Jianwei Chen
    • 3
    • 4
  • Jin-Ting Zhang
    • 5
    Email author
  1. 1.College of Economics and ManagementZhejiang University of TechnologyHangzhouChina
  2. 2.School of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouChina
  3. 3.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA
  4. 4.Center of Modern Applied Statistics and Big Data, School of StatisticsHuaqiao UniversityXiamenChina
  5. 5.Department of Statistics and Applied ProbabilityNational University of SingaporeSingaporeSingapore

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