Abstract

It is of great interest to test the equality of the means in two samples of functional data. Past research has predominantly concentrated on low-dimensional functional data, a focus that may not hold in high-dimensional scenarios. In this article, we propose a novel two-sample test for the mean functions of high-dimensional functional data, employing a multi-resolution projection (MRP) method. We establish the asymptotic normality of the proposed MRP test statistic and investigate its power performance when the dimension of the functional variables is high. In practice, functional data are observed only at discrete and usually asynchronous points. We further explore the influence of function reconstruction on the test statistic theoretically. Finally, we assess the finite-sample performance of the proposed test through extensive simulation studies and demonstrate its practicality via two real data applications. Specifically, our analysis of global climate data uncovers significant differences in the functional means of climate variables in the years 2020-2069 when comparing intermediate greenhouse gas emission pathways (e.g., RCP4.5) to high greenhouse gas emission pathways (e.g., RCP8.5).

Key words and phrases: Global climate data, High-dimensional functional data, Two-sample test, Mean function, Multi-resolution projection

Information

Preprint No.SS-2025-0356
Manuscript IDSS-2025-0356
Complete AuthorsShouxia Wang, Jiguo Cao, Hua Liu, Jinhong You, Jicai Liu
Corresponding AuthorsJicai Liu
Emailsliujicai1234@126.com

References

  1. Bai, Z. D. and H. Saranadasa (1996). Effect of high dimension: By an example of a two sample problem. Statistica Sinica 6(2), 311–329.
  2. Cai, T., W. Liu, and Y. Xia (2014). Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society Series B 76(2), 349–372.
  3. Chen, S. X. and Y. L. Qin (2010). A two-sample test for high-dimensional data with applications to gene-set testing. The Annals of Statistics 38(2), 808–835.
  4. Chou, S. C., A. Lyra, C. Mour˜ao, C. Dereczynski, I. Pilotto, J. Gomes, J. Bustamante,
  5. P. Tavares, A. Silva, and D. Rodrigues (2014). Assessment of climate change over South America under RCP 4.5 and 8.5 downscaling scenarios. American Journal of Climate Change 3(05), 512.
  6. Cuevas, A., M. Febrero, and R. Fraiman (2004). An ANOVA test for functional data. Computational Statistics & Data Analysis 47(1), 111–122.
  7. Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory 22(6), 1030–1051.
  8. Ferraty, F. and P. Vieu (2006). Nonparametric Functional Data Analysis: Theory and Practice. New York: Springer.
  9. Hall, P. and C. C. Heyde (1980). Martingale Limit Theory and its Application. Probability and Mathematical Statistics. New York: Academic Press.
  10. Horv´ath, L. and P. Kokoszka (2012). Inference for Functional Data with Applications. New York: Springer.
  11. Horv´ath, L., P. Kokoszka, and R. Reeder (2013). Estimation of the mean of functional time series and a two-sample problem. Journal of the Royal Statistical Society Series B 75(1), 103–122.
  12. Hsing, T. and R. Eubank (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Chichester: John Wiley & Sons.
  13. Jiang, Y., X. Wang, C. Wen, Y. Jiang, and H. Zhang (2024). Nonparametric two-sample tests of high dimensional mean vectors via random integration. Journal of the American Statistical Association 119(545), 701–714.
  14. Kim, I., S. Balakrishnan, and L. Wasserman (2020). Robust multivariate nonparametric tests via projection averaging. The Annals of Statistics 48(6), 3417–3441.
  15. Kokoszka, P. and M. Reimherr (2013). Determining the order of the functional autoregressive model: Order of functional autoregression. Journal of Time Series Analysis 34, 116–129.
  16. Liu, W., X. Yu, and R. Li (2022). Multiple-splitting projection test for high-dimensional mean vectors. Journal of Machine Learning Research 23(71), 1–27.
  17. Meinshausen, M., S. J. Smith, K. Calvin, J. S. Daniel, M. L. Kainuma, J.-F. Lamarque, K. Matsumoto, S. A. Montzka, S. C. Raper, and K. Riahi (2011). The RCP greenhouse gas concentrations and their extensions from 1765 to 2300. Climatic Change 109, 213–241.
  18. Moss, R. H., J. Edmonds, K. A. Hibbard, M. R. Manning, S. K. Rose, D. P. van Vuuren, T. R.
  19. Carter, S. Emori, M. Kainuma, T. Kram, G. A. Meehl, J. F. B. Mitchell, N. Nakicenovic, K. Riahi, S. J. Smith, R. J. Stouffer, A. M. Thomson, J. P. Weyant, and T. J. Wilbanks
  20. (2010). The next generation of scenarios for climate change research and assessment. Nature 463, 747–756.
  21. Qiu, Z., J. Chen, and J. Zhang (2021, May). Two-sample tests for multivariate functional data with applications. Computational Statistics & Data Analysis 157, 107160.
  22. Ramsay, J. O. and B. W. Silverman (2005). Functional Data Analysis. Springer, New York.
  23. Screen, J. A. and I. Simmonds (2010). The central role of diminishing sea ice in recent arctic temperature amplification. Nature 464(7293), 1334–1337.
  24. Tang, C., H. L. Shang, and Y. Yang (2022). Clustering and forecasting multiple functional time series. The Annals of Applied Statistics 16(4), 2523–2553.
  25. Thomson, A. M., K. V. Calvin, S. J. Smith, G. P. Kyle, A. Volke, P. Patel, S. DelgadoArias, B. Bond-Lamberty, M. A. Wise, and L. E. Clarke (2011). RCP4.5: a pathway for stabilization of radiative forcing by 2100. Climatic Change 109, 77–94.
  26. Wang, J.-L., J.-M. Chiou, and H.-G. M¨uller (2016). Functional data analysis. Annual Review of Statistics and Its Application 3, 257–295.
  27. Wynne, G. and A. B. Duncan (2022). A kernel two-sample test for functional data. Journal of Machine Learning Research 23(73), 1–51.
  28. Zhang, J. (2013). Analysis of Variance for Functional Data. Chapman and Hall, New York.
  29. Zhang, J. and X. Liang (2014). One-way ANOVA for functional data via globalizing the pointwise F-test. Scandinavian Journal of Statistics 41(1), 51–71.
  30. Zhu, C. and J. L. Wang (2023). Testing homogeneity: the trouble with sparse functional data. Journal of the Royal Statistical Society Series B 85(3), 705–731. Shouxia Wang, School of Statistics and Data Science, Shanghai University of Finance and

Acknowledgments

Shouxia Wang’s research was supported by National Natural Science Foundation of China (NSFC) (No.72501165). Jiguo Cao’s research was supported by Natural Sciences and Engineering Research Council of Canada Discovery grant (RGPIN-2023-04057). Hua Liu’s research was supported by the NSFC (No.12201487). Jinhong You’s research was supported by the 111-Center Project of China (No. B25066). The corresponding authors are Jicai Liu and Jinhong You (co-corresponding).

Supplementary Materials

We provide additional simulation and real data analysis results, and technical details in the Supplementary Material.

Supplementary materials are available for download.