Electronic Journal of Statistics

Simultaneous inference of the mean of functional time series

Ming Chen and Qiongxia Song

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For functional time series with physical dependence, we construct confidence bands for its mean function. The physical dependence is a general dependence framework, and it slightly relaxes the conditions of m-approximable dependence. We estimate functional time series mean functions via a spline smoothing technique. Confidence bands have been constructed based on a long-run variance and a strong approximation theorem, which is satisfied with mild regularity conditions. Simulation experiments provide strong evidence that corroborates the asymptotic theories. Additionally, an application to S&P500 index data demonstrates a non-constant volatility mean function at a certain significance level.

Article information

Electron. J. Statist., Volume 9, Number 2 (2015), 1779-1798.

Received: April 2015
First available in Project Euclid: 25 August 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G15: Tolerance and confidence regions

Confidence bands functional time series high-frequency data long-run variance nonparametric regression spline


Chen, Ming; Song, Qiongxia. Simultaneous inference of the mean of functional time series. Electron. J. Statist. 9 (2015), no. 2, 1779--1798. doi:10.1214/15-EJS1052. https://projecteuclid.org/euclid.ejs/1440507393

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  • [1] Aue, A., Norinho, D. D., and Hörmann, S. (2012). On the prediction of functional time series., arXiv:1208.2892.
  • [2] Berkes, I., Liu, W., and Wu, W. B. (2014). Komlós-Major-Tusnády approximation under dependence., The Annals of Probability, 42(2):794–817.
  • [3] Bosq, D. (2000)., Linear Processes in Function Spaces: Theory and Applications, volume 149. Springer.
  • [4] Cao, G., Yang, L., and Todem, D. (2012). Simultaneous inference for the mean function based on dense functional data., Journal of Nonparametric Statistics, 24(2):359–377.
  • [5] Castro, P. E., Lawton, W. H., and Sylvestre, E. A. (1986). Principal modes of variation for processes with continuous sample curves., Technometrics, 28(4):329–337.
  • [6] Claeskens, G. and Keilegom, I. v. (2003). Bootstrap confidence bands for regression curves and their derivatives., The Annals of Statistics, 31(6):1852–1884.
  • [7] de Boor, C. (2001)., A Practical Guide to Splines. Springer-Verlag, Berlin.
  • [8] Evans, K. P. and Speight, A. E. (2010). Intraday periodicity, calendar and announcement effects in euro exchange rate volatility., Research in International Business and Finance, 24(1):82–101.
  • [9] Fan, J. and Zhang, W. (2000). Simultaneous confidence bands and hypothesis testing in varying-coefficient models., Scandinavian Journal of Statistics, 27(4):715–731.
  • [10] Ferraty, F. and Vieu, P. (2006)., Nonparametric Functional Data Analysis: Theory and Practice. Springer.
  • [11] Hall, P. and Titterington, D. (1988). On confidence bands in nonparametric density estimation and regression., Journal of Multivariate Analysis, 27(1):228–254.
  • [12] Hörmann, S., Horváth, L., and Reeder, R. (2013). A functional version of the arch model., Econometric Theory, 29:267–288.
  • [13] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data., The Annals of Statistics, 38(3):1845–1884.
  • [14] Horváth, L., Kokoszka, P., and Reeder, R. (2013). Estimation of the mean of functional time series and a two-sample problem., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(1):103–122.
  • [15] Liu, W. and Lin, Z. (2009). Strong approximation for a class of stationary processes., Stochastic Processes and Their Applications, 119(1):249–280.
  • [16] Ma, S., Yang, L., and Carroll, R. J. (2012). A simultaneous confidence band for sparse longitudinal regression., Statistica Sinica, 22:95–122.
  • [17] Müller, H.-G., Stadtmüller, U., and Yao, F. (2006). Functional variance processes., Journal of the American Statistical Association, 101(475):1007–1018.
  • [18] Ramsay, J. O. and Silverman, B. (2005)., Functional Data Analysis. Springer, 2nd edition.
  • [19] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves., Journal of the Royal Statistical Society. Series B (Methodological), 53(1):233–243.
  • [20] Wang, J. and Yang, L. (2009). Polynomial spline confidence bands for regression curves., Statistica Sinica, 19(1):325–342.
  • [21] Yao, F. (2007). Functional principal component analysis for longitudinal and survival data., Statistica Sinica, 17(3):965–983.
  • [22] Yao, F., Müller, H.-G., and Wang, J.-L. (2005a). Functional data analysis for sparse longitudinal data., Journal of the American Statistical Association, 100(470):577–590.
  • [23] Yao, F., Müller, H.-G., and Wang, J.-L. (2005b). Functional linear regression analysis for longitudinal data., The Annals of Statistics, 33(6):2873–2903.
  • [24] Zhao, Z. and Wu, W. B. (2008). Confidence bands in nonparametric time series regression., The Annals of Statistics, 36(4):1854–1878.
  • [25] Zhou, S., Shen, X., and Wolfe, D. (1998). Local asymptotics for regression splines and confidence regions., The Annals of Statistics, 26(5):1760–1782.