Bernoulli

  • Bernoulli
  • Volume 21, Number 2 (2015), 909-929.

Two sample inference for the second-order property of temporally dependent functional data

Xianyang Zhang and Xiaofeng Shao

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Abstract

Motivated by the need to statistically quantify the difference between two spatio-temporal datasets that arise in climate downscaling studies, we propose new tests to detect the differences of the covariance operators and their associated characteristics of two functional time series. Our two sample tests are constructed on the basis of functional principal component analysis and self-normalization, the latter of which is a new studentization technique recently developed for the inference of a univariate time series. Compared to the existing tests, our SN-based tests allow for weak dependence within each sample and it is robust to the dependence between the two samples in the case of equal sample sizes. Asymptotic properties of the SN-based test statistics are derived under both the null and local alternatives. Through extensive simulations, our SN-based tests are shown to outperform existing alternatives in size and their powers are found to be respectable. The tests are then applied to the gridded climate model outputs and interpolated observations to detect the difference in their spatial dynamics.

Article information

Source
Bernoulli, Volume 21, Number 2 (2015), 909-929.

Dates
First available in Project Euclid: 21 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1429624965

Digital Object Identifier
doi:10.3150/13-BEJ592

Mathematical Reviews number (MathSciNet)
MR3338651

Zentralblatt MATH identifier
06445962

Keywords
climate downscaling functional data analysis long run variance matrix self-normalization time series two sample problem

Citation

Zhang, Xianyang; Shao, Xiaofeng. Two sample inference for the second-order property of temporally dependent functional data. Bernoulli 21 (2015), no. 2, 909--929. doi:10.3150/13-BEJ592. https://projecteuclid.org/euclid.bj/1429624965


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References

  • [1] Benko, M., Härdle, W. and Kneip, A. (2009). Common functional principal components. Ann. Statist. 37 1–34.
  • [2] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. New York: Wiley.
  • [3] Bosq, D. (2000). Linear Processes in Function Spaces. Lecture Notes in Statistics 149. New York: Springer. Theory and applications.
  • [4] Cuevas, A., Febrero, M. and Fraiman, R. (2004). An ANOVA test for functional data. Comput. Statist. Data Anal. 47 111–122.
  • [5] Fan, J. and Lin, S.-K. (1998). Test of significance when data are curves. J. Amer. Statist. Assoc. 93 1007–1021.
  • [6] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis. Theory and Practice. Springer Series in Statistics. New York: Springer.
  • [7] Fremdt, S., Steinebach, J.G., Horváth, L. and Kokoszka, P. (2013). Testing the equality of covariance operators in functional samples. Scand. J. Stat. 40 138–152.
  • [8] Gabrys, R. and Kokoszka, P. (2007). Portmanteau test of independence for functional observations. J. Amer. Statist. Assoc. 102 1338–1348.
  • [9] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38 1845–1884.
  • [10] Horváth, L., Kokoszka, P. and Reeder, R. (2013). Estimation of the mean of functional time series and a two-sample problem. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 103–122.
  • [11] Jun, M., Knutti, R. and Nychka, D.W. (2008). Spatial analysis to quantify numerical model bias and dependence: How many climate models are there? J. Amer. Statist. Assoc. 103 934–947.
  • [12] Kraus, D. and Panaretos, V.M. (2012). Dispersion operators and resistant second-order functional data analysis. Biometrika 99 813–832.
  • [13] Lahiri, S.N. (2003). Resampling Methods for Dependent Data. Springer Series in Statistics. New York: Springer.
  • [14] Lobato, I.N. (2001). Testing that a dependent process is uncorrelated. J. Amer. Statist. Assoc. 96 1066–1076.
  • [15] McMurry, T. and Politis, D. (2011). Resampling methods for functional data. In The Oxford Handbook of Functional Data Analysis (F. Ferraty and Y. Romain, eds.) 189–209. Oxford Univ. Press, Oxford.
  • [16] Panaretos, V.M., Kraus, D. and Maddocks, J.H. (2010). Second-order comparison of Gaussian random functions and the geometry of DNA minicircles. J. Amer. Statist. Assoc. 105 670–682.
  • [17] Politis, D.N. and Romano, J.P. (2010). $K$-sample subsampling in general spaces: The case of independent time series. J. Multivariate Anal. 101 316–326.
  • [18] Politis, D.N., Romano, J.P. and Wolf, M. (1999). Subsampling. Springer Series in Statistics. New York: Springer.
  • [19] Ramsay, J.O. and Silverman, B.W. (2002). Applied Functional Data Analysis. Springer Series in Statistics. New York: Springer. Methods and case studies.
  • [20] Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [21] Shao, X. (2010). A self-normalized approach to confidence interval construction in time series. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 343–366.
  • [22] Zhang, X. and Shao, X. (2014). Supplement to “Two sample inference for the second-order property of temporally dependent functional data.” DOI:10.3150/13-BEJ592SUPP.
  • [23] Zhang, X., Shao, X., Hayhoe, K. and Wuebbles, D.J. (2011). Testing the structural stability of temporally dependent functional observations and application to climate projections. Electron. J. Stat. 5 1765–1796.

Supplemental materials

  • Supplementary material: Supplement to “Two sample inference for the second-order property of temporally dependent functional data”. This supplement contains proofs of the results in Section 3.