Electronic Journal of Statistics

Simultaneous confidence bands for derivatives of dependent functional data

Guanqun Cao

Full-text: Open access


In this work, consistent estimators and simultaneous confidence bands for the derivatives of mean functions are proposed when curves are repeatedly recorded for each subject. The within-curve correlation of trajectories has been considered while the proposed novel confidence bands still enjoys semiparametric efficiency. The proposed methods lead to a straightforward extension of the two-sample case in which we compare the derivatives of mean functions from two populations. We demonstrate in simulations that the proposed confidence bands are superior to existing approaches which ignore the within-curve dependence. The proposed methods are applied to investigate the derivatives of mortality rates from period lifetables that are repeatedly collected over many years for various countries.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 2639-2663.

First available in Project Euclid: 22 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G10: Hypothesis testing 62G15: Tolerance and confidence regions 62G20: Asymptotic properties

B-spline confidence band derivatives functional data semiparametric efficiency repeated measures


Cao, Guanqun. Simultaneous confidence bands for derivatives of dependent functional data. Electron. J. Statist. 8 (2014), no. 2, 2639--2663. doi:10.1214/14-EJS967. https://projecteuclid.org/euclid.ejs/1419258189

Export citation


  • [1] Bunea, F., Ivanescu, A.E. and Wegkamp, M. (2011), “Adaptive inference for the mean of a Gaussian process in functional data,”, Journal of the Royal Statistical Society, Series B, 73, 531–558.
  • [2] Cao, G. and Wang, L. (2014), “Simultaneous inference for repeated functional data,”, Manuscript.
  • [3] Cao, G., Wang, L., Wang, J. and Todem, D. (2012), “Spline confidence bands for functional derivatives,”, Journal of Statistical Planning and Inference, 142, 1557–1570.
  • [4] Cao, G., Yang, L. and Todem, D. (2012), “Simultaneous inference for the mean function of dense functional data,”, Journal of Nonparametric Statistics, 24, 359–377.
  • [5] Chen, K. and Müller, H.G. (2012), “Modeling repeated functional observations,”, Journal of the American Statistical Association, 107, 1599–1609.
  • [6] Chiou, J.M. and Müller, H.G. (2009), “Modeling hazard rates as functional data for the analysis of cohort lifetables and mortality forecasting,”, Journal of the American Statistical Association, 104, 572–585.
  • [7] de Boor, C. (2001), A Practical Guide to Splines, New York: Springer-Verlag.
  • [8] Degras, D.A. (2011), “Simultaneous confidence bands for nonparametric regression with functional data,”, Statistica Sinica, 21, 1735–1765.
  • [9] Di, C., Crainiceanu, C.M., Caffo, B.S. and Punjabi, N.M. (2009), “Multilevel functional principal component Models,”, Annals of Applied Statistics, 3, 458–488.
  • [10] DeVore, R. and Lorentz, G. (1993), Constructive Approximation: Polynomials and Splines Approximation, Berlin: Springer-Verlag.
  • [11] Ferraty, F. and Vieu, P. (2006), NonParametric Functional Data Analysis: Theory and Practice, New York: Springer Series in Statistics, Springer.
  • [12], Human Mortality Database, University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (data download on January 24, 2013 and May 9, 2013).
  • [13] Liang, K.Y. and Zeger, S.L. (1986), “Longitudinal data analysis using generalized linear models,”, Biometrika, 73, 13–22.
  • [14] Liu, B. and Müller, H.G. (2009), “Estimating derivatives for samples of sparsely observed functions, with application to on-line auction dynamics,”, Journal of American Statistical Association, 104, 704–717.
  • [15] Ramsay, J.O. and Silverman, B.W. (2005), Functional Data Analysis, Second Edition, New York: Springer Series in Statistics, Springer.
  • [16] Wang, J. and Yang, L. (2009), “Polynomial spline confidence bands for regression curves,”, Statistica Sinica, 19, 325–342.
  • [17] Yao, F., Müller, H.G. and Wang, J.L. (2005), “Functional data analysis for sparse longitudinal data,”, Journal of the American Statistical Association, 100, 577–590.
  • [18] Yuan, Y., Gilmore, J.H., Geng, X., Martin, S., Chen, K., Wang, J.L. and Zhu, H. (2014), “FMEM: functional mixed effects modeling for the analysis of longitudinal white matter tract data,”, Neuroimage, 84, 753–764.
  • [19] Zhou, S. and Wolfe, D. A. (2000), “On derivative estimation in spline regression,”, Statistica Sinica 10, 93–108.
  • [20] Zhu, H., Li, R. and Kong, L. (2012), “Multivariate varying coefficient model for functional responses,”, Annals of Statistics, 40, 2634–2666.