Skip to main content
SpringerLink
Log in

Distribution/correlation-free test for two-sample means in high-dimensional functional data with eigenvalue decay relaxed

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We propose a methodology for testing two-sample means in high-dimensional functional data that requires no decaying pattern on eigenvalues of the functional data. To the best of our knowledge, we are the first to consider such a problem and address it. To be specific, we devise a confidence region for the mean curve difference between two samples, which directly establishes a rigorous inferential procedure based on the multiplier bootstrap. In addition, the proposed test permits the functional observations in each sample to have mutually different distributions and arbitrary correlation structures, which is regarded as the desired property of distribution/correlation-free, leading to a more challenging scenario for theoretical development. Other desired properties include the allowance for highly unequal sample sizes, exponentially growing data dimension in sample sizes and consistent power behavior under fairly general alternatives. The proposed test is shown uniformly convergent to the prescribed significance, and its finite sample performance is evaluated via the simulation study and an implementation to electroencephalography data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Bai Z D, Saranadasa H. Effect of high dimension: By an example of a two sample problem. Statist Sinica, 1996, 6: 311–329

    MathSciNet  MATH  Google Scholar 

  2. Cai T T, Hall P. Prediction in functional linear regression. Ann Statist, 2006, 34: 2159–2179

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai T T, Liu W D, Xia Y. Two-sample test of high dimensional means under dependence. J R Stat Soc Ser B Stat Methodol, 2014, 76: 349–372

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang J Y, Zheng C, Zhou W-X, et al. Simulation-based hypothesis testing of high dimensional means under covariance heterogeneity. Biometrics, 2017, 73: 1300–1310

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen S X, Qin Y-L. A two-sample test for high-dimensional data with applications to gene-set testing. Ann Statist, 2010, 38: 808–835

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan Y Y, James G M, Radchenko P. Functional additive regression. Ann Statist, 2015, 43: 2296–2325

    Article  MathSciNet  MATH  Google Scholar 

  7. Gregory K B, Carroll R J, Baladandayuthapani V, et al. A two-sample test for equality of means in high dimension. J Amer Statist Assoc, 2015, 110: 837–849

    Article  MathSciNet  MATH  Google Scholar 

  8. Hall P, Horowitz J L. Methodology and convergence rates for functional linear regression. Ann Statist, 2007, 35: 70–91

    Article  MathSciNet  MATH  Google Scholar 

  9. Hall P, Hosseini-Nasab M. On properties of functional principal components analysis. J R Stat Soc Ser B Stat Methodol, 2006, 68: 109–126

    Article  MathSciNet  MATH  Google Scholar 

  10. Hall P, van Keilegom I. Two-sample tests in functional data analysis starting from discrete data. Statist Sinica, 2007, 17: 1511–1531

    MathSciNet  MATH  Google Scholar 

  11. Hussain L, Aziz W, Nadeem S A, et al. Electroencephalography (EEG) analysis of alcoholic and control subjects using multiscale permutation entropy. J Multidiscip Engrg Sci Technol, 2014, 1: 380–387

    Google Scholar 

  12. Kong D H, Xue K J, Yao F, et al. Partially functional linear regression in high dimensions. Biometrika, 2016, 103: 147–159

    Article  MathSciNet  MATH  Google Scholar 

  13. Krzyśko M, Smaga L. Two-sample tests for functional data using characteristic functions. Austrian J Statist, 2021, 50: 53–64

    Article  Google Scholar 

  14. Lee J S, Cox D D, Follen M. A two sample test for functional data. Comm Statist Appl Methods, 2015, 22: 121–135

    Article  Google Scholar 

  15. Lin Z H, Lopes M E, Müller H-G. High-dimensional MANOVA via bootstrapping and its application to functional and sparse count data. J Amer Statist Assoc, 2023, in press

  16. Pomann G-M, Staicu A-M, Ghosh S. A two-sample distribution-free test for functional data with application to a diffusion tensor imaging study of multiple sclerosis. J R Stat Soc Ser C Appl Stat, 2016, 65: 395–414

    Article  MathSciNet  Google Scholar 

  17. Reiss P T, Ogden R T. Functional principal component regression and functional partial least squares. J Amer Statist Assoc, 2007, 102: 984–996

    Article  MathSciNet  MATH  Google Scholar 

  18. Rice J A, Silverman B W. Estimating the mean and covariance structure nonparametrically when the data are curves. J R Stat Soc Ser B Stat Methodol, 1991, 53: 233–243

    MathSciNet  MATH  Google Scholar 

  19. Srivastava M S, Kubokawa T. Tests for multivariate analysis of variance in high dimension under non-normality. J Multivariate Anal, 2013, 115: 204–216

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang Q Y. Two-sample inference for sparse functional data. Electron J Stat, 2021, 15: 1395–1423

    Article  MathSciNet  MATH  Google Scholar 

  21. Xue K J, Yao F. Distribution and correlation-free two-sample test of high-dimensional means. Ann Statist, 2020, 48: 1304–1328

    Article  MathSciNet  MATH  Google Scholar 

  22. Xue K J, Yao F. Hypothesis testing in large-scale functional linear regression. Statistica Sinica, 2021, 31: 1101–1123

    MathSciNet  MATH  Google Scholar 

  23. Yao F, Müller H-G, Wang J-L. Functional linear regression analysis for longitudinal data. Ann Statist, 2005, 33: 2873–2903

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang C Q, Peng H, Zhang J-T. Two samples tests for functional data. Comm Statist Theory Methods, 2010, 39: 559–578

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11901313), Fundamental Research Funds for the Central Universities, Key Laboratory for Medical Data Analysis and Statistical Research of Tianjin, and Key Laboratory of Pure Mathematics and Combinatorics, Ministry of Education. The author thanks the referees for their insightful comments.

Author information

Author notes

  1. Kaijie Xue

    Present address: Current address: School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai, 201620, China

Authors and Affiliations

  1. School of Statistics and Data Science, Nankai University, Tianjin, 300071, China

    Kaijie Xue

Authors
  1. Kaijie Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kaijie Xue.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, K. Distribution/correlation-free test for two-sample means in high-dimensional functional data with eigenvalue decay relaxed. Sci. China Math. 66, 2337–2346 (2023). https://doi.org/10.1007/s11425-022-2042-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-022-2042-6

Keywords

MSC(2020)

Search

Navigation