Abstract
The permutation approach for testing the equality of distributions and thereby comparing two populations of functional data has recently received increasing attention thanks to the flexibility of permutation tests to handle complex testing problems. The purpose of this work is to present some new insights in the context of nonparametric inference on functional data using the permutation approach, more specifically we formally show the equivalence of some permutation procedures proposed in the literature and we suggest the use of the permutation and combination-based approach within the basis function approximation layout. Validation of theoretical results is shown by simulation studies.
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References
Bai Z, Saranadasa H (1996) Effect of high dimension: by an example of a two sample problem. Statistica Sinica 6(2):311–329
Basso D, Pesarin F, Salmaso L, Solari A (2009) Permutation tests for Stochastic ordering and ANOVA: theory and applications with R. In: Lecture notes in Statistics, Springer, New York, USA
Brunner E, Bathke A, Placzek M (2012) Estimation of box’s \(epsilon\) for low- and high-dimensional repeated measures designs with unequal covariance matrices. Biometr J 54(3):301–316
Cao G, Yang L, Todem D (2012) Simultaneous inference for the mean function based on dense functional data. J Nonparametr Stat 24:359–377
Cardot H, Prchal L, Sarda P (2007) No effect and lack-of-fit permutation tests for functional regression. Comput Stat 22:371–390
Corain L, Melas V, Salmaso L, Pepelyshev A (2013) Optimal adaptive Fourier-Based testing for the equality of two nonparametric regression curves. Biometr J
Cox D, Lee J (2008) Pointwise testing with functional data using the Westfall-Young randomization method. Biometrika 95:621–634
Cuevas A, Febrero M, Fraiman R (2004) An ANOVA test for functional data. Comput Stat Data Anal 47:111–122
Davidian M, Lin X, Wang J (2004) Introduction: emerging issues in longitudinal and functional data analysis. Stat Sinica 14:613–614
Diggle P, Heagerty P, Liang K-Y, Zeger S (2002) Analysis of longitudinal data. In: 2nd edn. Oxford University Press
Edgington E, Onghena P (2007) Randomization tests, 4th edn. Chapman and Hall, London
Fan J, Li S-K (1998) Test of significance when data are curves. J Am Stat Assoc 93:1007–1021
Ferraty F, Vieu P (2006) Nonparametric functional data analysis., Springer series in statistics Springer, New York
Folks J (1984) Combinations of independent tests. In: Krishnaiah P, Sen P (eds) Handbook of statistics, vol 4. North-Holland, Amsterdam, pp 113–121
Good P (2010) Permutation, parametric, and bootstrap tests of hypotheses, 3rd edn., Springer series in statistics Springer, New York
Hall P, Van Keilegom I (2007) Two-sample tests in functional data analysis starting from discrete data. Stat Sinica 17:1511–1531
Konietschke F, Pauly M (2013) Bootstrapping and permuting paired t-test type statistics. Stat Comput. doi:10.1007/s11222-012-9370-4
Martinez-Camblor P, Corral N (2011) Repeated measures analysis for functional data. Comput Stat Data Anal 55:3244–3256
Mielke P, Berry K (2007) Permutation methods, a distance function approach, 2nd edn. Springer, New York
Mohdeb Z, Mezhoud K, Boudaa D (2010) Testing the equality of nonparametric regression curves based on Fourier coefficients. J Afrika Stat 5:219–227
Munoz-Maldonado Y, Staniswalis J, Irwin L, Byers D (2002) A similarity analysis of curves. Can J Stat 30:373–381
Pesarin F, Salmaso L (2010) Permutation tests for complex data: theory. Applications and software, Wiley series in probability and statistics, Chichester, UK
Ramsay J, Hooker G, Graves S (2009) Functional data analysis with R and Matlab. Springer, New York
Ramsay J, Silverman B (2005) Functional data analysis, 2nd edn. Springer, New York
Reboussin D, DeMets D (1996) Exact permutation inference for two sample repeated measures data. Commun Stat Theory Methods 25:2223–2238
Rice J (2004) Functional and longitudinal data analysis: perspectives on smoothing. Stat Sinica 14:631–647
Shen Q, Faraway J (2004) An F test for linear models with functional responses. Stat Sinica 14:1239–1257
Sirski M (2012) On the statistical analysis of functional data arising from designed experiments. Ph.D. Thesis, Department of Statistics, University of Manitoba
Spitzner D (2008) A powerful test based on tapering for use in functional data analysis. Electron J Stat 2:939–962
Sturino J, Zorych I, Mallick B, Pokusaeva K, Chang Y-Y, Carroll R, Bliznuyk N (2010) Statistical methods for comparative phenomics using high-throughput phenotype microarrays. Int J Biostat 6
Wang H, Akritas M (2011) Asymptotically distribution free tests in heteroscedastic unbalanced high dimensional anova. Stat Sinica 21:1341–1377
Zhang J-T, Chen J (2007) Statistical inferences for functional data. Annal Stat 35:1052–1079
Acknowledgments
This work was elaborated during the visit of Viatcheslav Melas at the Department of Management and Engineering, University of Padova, as a Visiting Scientist in July 2012. The work of Viatcheslav Melas and Andrey Pepelyshev was supported in part by Russian Foundation of Basic Research, project 12-01-00747. Authors wish to thank Fortunato Pesarin for stimulating discussions on topics related to this paper. Also we wish to thank the anonymous referees for very useful remarks that allowed us to improve the paper. This paper has to be attributed in equal parts to all co-authors, as a result of a full collaboration among all co-authors.
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Corain, L., Melas, V.B., Pepelyshev, A. et al. New insights on permutation approach for hypothesis testing on functional data. Adv Data Anal Classif 8, 339–356 (2014). https://doi.org/10.1007/s11634-013-0162-2
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DOI: https://doi.org/10.1007/s11634-013-0162-2
Keywords
- Data compression
- Functional data
- Nonparametric combination
- Permutation tests
- Permutationally equivalent test statistic