Advertisement

M-based simultaneous inference for the mean function of functional data

  • Italo R. Lima
  • Guanqun CaoEmail author
  • Nedret Billor
Article
  • 263 Downloads

Abstract

Estimating and constructing a simultaneous confidence band for the mean function in the presence of outliers is an important problem in the framework of functional data analysis. In this paper, we propose a robust estimator and a robust simultaneous confidence band for the mean function of functional data using M-estimation and B-splines. The robust simultaneous confidence band is also extended to the difference of mean functions of two populations. Further, the asymptotic properties of the M-based mean function estimator, such as the asymptotic consistency and asymptotic normality, are studied. The performance of the proposed robust methods and their robustness are demonstrated with an extensive simulation study and two real data examples.

Keywords

Confidence band Functional data analysis Robust statistics Spline smoothing M-estimator Pseudo-data 

Supplementary material

10463_2018_656_MOESM1_ESM.pdf (250 kb)
Supplementary material 1 (pdf 249 KB)

References

  1. Bali, J. L., Boente, G., Tyler, D. E., Wang, J. L. (2011). Robust functional principal components: A projection-pursuit approach. Annals of Statistics, 39(6), 2852–2882.Google Scholar
  2. Boente, G., Salibian-Barrera, M. (2015). S-estimators for functional principal component analysis. Journal of the American Statistical Association, 110(511), 1100–1111.Google Scholar
  3. Cao, G., Yang, L., Todem, D. (2012). Simultaneous inference for the mean function based on dense functional data. Journal of Nonparametric Statistics, 24(2), 359–377.Google Scholar
  4. Cox, D. D. (1983). Asymptotics for m-type smoothing splines. Annals of Statistics, 11, 530–551.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Daszykowski, M., Kaczmarek, K., Vander Heyden, Y., Walczak, B. (2007). Robust statistics in data analysis—A review: Basic concepts. Chemometrics and Intelligent Laboratory Systems, 85(2), 203–219.Google Scholar
  6. Embling, C. B., Illian, J., Armstrong, E., van der Kooij, J., Sharples, J., Camphuysen, K. C., Scott, B. E. (2012). Investigating fine-scale spatio-temporal predator-prey patterns in dynamic marine ecosystems: A functional data analysis approach. Journal of Applied Ecology, 49(2), 481–492.Google Scholar
  7. Esbensen, K., Schönkopf, S., Midtgaard, T., Guyot, D. (1996). Multivariate analysis in practice: A training package. Trondheim: Camo As.Google Scholar
  8. Febrero, M., Galeano, P., González-Manteiga, W. (2008). Outlier detection in functional data by depth measures, with application to identify abnormal nox levels. Environmetrics, 19(4), 331–345.Google Scholar
  9. Ferraty, F. (2011). Recent advances in functional data analysis and related topics. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  10. Ferraty, F., Rabhi, A., Vieu, P. (2005). Conditional quantiles for dependent functional data with application to the climatic “el niño” phenomenon. Sankhyā: The Indian Journal of Statistics, 67(2), 378–398.Google Scholar
  11. Gervini, D. (2008). Robust functional estimation using the median and spherical principal components. Biometrika, 95(3), 587–600.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gu, L., Wang, L., Härdle, W. K., Yang, L. (2014). A simultaneous confidence corridor for varying coefficient regression with sparse functional data. Test, 23(4), 806–843.Google Scholar
  13. Huang, J. Z., Wu, C. O., Zhou, L. (2004). Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Statistica Sinica, 14, 763–788.Google Scholar
  14. Huber, P. J. (1964). Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1), 73–101.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kraus, D., Panaretos, V. M. (2012). Dispersion operators and resistant second-order functional data analysis. Biometrika, 99(4), 813–832.Google Scholar
  16. Lee, S., Shin, H., Billor, N. (2013). M-type smoothing spline estimators for principal functions. Computational Statistics & Data Analysis, 66, 89–100.Google Scholar
  17. Lim, Y., Oh, H. S. (2015). Simultaneous confidence interval for quantile regression. Computational Statistics, 30(2), 345–358.Google Scholar
  18. Lima, I. R., Cao, G., Billor, N. (2017). Robust simultaneous inference for the mean function of functional data. Ph.D. dissertation. Auburn University.Google Scholar
  19. Locantore, N., Marron, J., Simpson, D., Tripoli, N., Zhang, J., Cohen, K., Boente, G., Fraiman, R., Brumback, B., Croux, C. (1999). Robust principal component analysis for functional data. Test, 8(1), 1–73.Google Scholar
  20. Maronna, R., Martin, D., Yohai, V. (2006). Robust statistics: Theory and methods. Wiley series in probability and statistics. Chichester: Wiley.Google Scholar
  21. Maronna, R. A., Yohai, V. J. (2013). Robust functional linear regression based on splines. Computational Statistics & Data Analysis, 65, 46–55.Google Scholar
  22. Shin, H., Lee, S. (2016). An RKHS approach to robust functional linear regression. Statistica Sinica, 26, 255–272.Google Scholar
  23. Silverman, B., Ramsay, J. (2005). Functional data analysis (2nd ed.). New York: Springer.Google Scholar
  24. Stone, C. J. (1985). Additive regression and other nonparametric models. The Annals of Statistics, 13, 689–705.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Tang, Q., Cheng, L. (2012). M-estimation and b-spline approximation for varying coefficient models with longitudinal data. Journal of Nonparametric Statistics, 20, 611–625.Google Scholar
  26. Venables, W. N., Ripley, B. D. (2002). Modern applied statistics with S (4th ed.). New York: Springer.Google Scholar
  27. Wei, Y., He, X. (2006). Conditional growth charts. Annals of Statistics, 34(5), 2069–2097.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

Personalised recommendations