Annals of Statistics

Optimal estimation of the mean function based on discretely sampled functional data: Phase transition

T. Tony Cai and Ming Yuan

Full-text: Open access


The problem of estimating the mean of random functions based on discretely sampled data arises naturally in functional data analysis. In this paper, we study optimal estimation of the mean function under both common and independent designs. Minimax rates of convergence are established and easily implementable rate-optimal estimators are introduced. The analysis reveals interesting and different phase transition phenomena in the two cases. Under the common design, the sampling frequency solely determines the optimal rate of convergence when it is relatively small and the sampling frequency has no effect on the optimal rate when it is large. On the other hand, under the independent design, the optimal rate of convergence is determined jointly by the sampling frequency and the number of curves when the sampling frequency is relatively small. When it is large, the sampling frequency has no effect on the optimal rate. Another interesting contrast between the two settings is that smoothing is necessary under the independent design, while, somewhat surprisingly, it is not essential under the common design.

Article information

Ann. Statist., Volume 39, Number 5 (2011), 2330-2355.

First available in Project Euclid: 30 November 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Functional data mean function minimax rate of convergence phase transition reproducing kernel Hilbert space smoothing splines Sobolev space


Cai, T. Tony; Yuan, Ming. Optimal estimation of the mean function based on discretely sampled functional data: Phase transition. Ann. Statist. 39 (2011), no. 5, 2330--2355. doi:10.1214/11-AOS898.

Export citation


  • Aronszajn, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 337–404.
  • Assouad, P. (1983). Deux remarques sur l’estimation. C. R. Acad. Sci. Paris Sér. I Math. 296 1021–1024.
  • Cai, T. and Yuan, M. (2010). Nonparametric covariance function estimation for functional and longitudinal data. Technical report, Georgia Institute of Technology, Atlanta, GA.
  • DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 303. Springer, Berlin.
  • Diggle, P., Heagerty, P., Liang, K. and Zeger, S. (2002). Analysis of Longitudinal Data, 2nd ed. Oxford Univ. Press, Oxford.
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York.
  • Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Monographs on Statistics and Applied Probability 58. Chapman and Hall, London.
  • Hall, P. and Hart, J. D. (1990). Nonparametric regression with long-range dependence. Stochastic Process. Appl. 36 339–351.
  • Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493–1517.
  • Hart, J. D. and Wehrly, T. E. (1986). Kernel regression estimation using repeated measurements data. J. Amer. Statist. Assoc. 81 1080–1088.
  • James, G. M. and Hastie, T. J. (2001). Functional linear discriminant analysis for irregularly sampled curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 533–550.
  • Johnstone, I. M. and Silverman, B. W. (1997). Wavelet threshold estimators for data with correlated noise. J. Roy. Statist. Soc. Ser. B 59 319–351.
  • Opsomer, J., Wang, Y. and Yang, Y. (2001). Nonparametric regression with correlated errors. Statist. Sci. 16 134–153.
  • Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer, New York.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243.
  • Rice, J. A. and Wu, C. O. (2001). Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics 57 253–259.
  • Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040–1053.
  • Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York.
  • Wahba, G. (1990). Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics 59. SIAM, Philadelphia, PA.
  • Wang, Y. (1996). Function estimation via wavelet shrinkage for long-memory data. Ann. Statist. 24 466–484.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.