A more powerful test identifying the change in mean of functional data

  • Buddhananda Banerjee
  • Satyaki Mazumder


An existence of change point in a sequence of temporally ordered functional data demands more attention in its statistical analysis to make a better use of it. Introducing a dynamic estimator of covariance kernel, we propose a new methodology for testing an existence of change in the mean of temporally ordered functional data. Though a similar estimator is used for the covariance in finite dimension, we introduce it for the independent and weakly dependent functional data in this context for the first time. From this viewpoint, the proposed estimator of covariance kernel is more natural one when the sequence of functional data may possess a change point. We prove that the proposed test statistics are asymptotically pivotal under the null hypothesis and consistent under the alternative. It is shown that our testing procedures outperform the existing ones in terms of power and provide satisfactory results when applied to real data.


Change point detection Functional data analysis Covariance kernel 



The authors are thankful to the British Atmospheric Data Centre and Carbon Dioxide Information Analysis Center for real data. The Daily Central England Temperature data have been taken from the Hadley Centre for Climate Prediction and Research (2007), and monthly global average anomaly of temperatures is taken from Jones et al. (2013). The authors also thank the anonymous referee for his insightful comments and suggestions which helped to make a significant improvement of the manuscript.


  1. Antoch, J., Husková, M., Prásková, Z. (1997). Effect of dependence on statistics for determination of change. Journal of Statistical Planning and Inference, 60, 291–310.Google Scholar
  2. Aston, J. A., Kirch, C. (2012). Detecting and estimating changes in dependent functional data. Journal of Multivariate Analysis, 109, 204–220.Google Scholar
  3. Aue, A., Gabrys, R., Horváth, L., Kokoszka, P. (2009). Estimation of a change-point in the mean function of functional data. Journal of Multivariate Analysis, 100(10), 2254–2269.Google Scholar
  4. Berkes, I., Gabrys, R., Horváth, L., Kokoszka, P. (2009). Detecting changes in the mean of functional observations. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(5), 927–946.Google Scholar
  5. Bhattacharya, P. K. (1987). Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multi-parameter case. Journal of Multivariate Analysis, 23, 183–208.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bongiorno, E. G., Salinelli, E., Goia, A., Vieu, P. (2014). Contributions in infinite-dimensional statistics and related topics. Chicago: Societa Editrice Esculapio.Google Scholar
  7. Bosq, D. (2000). Linear processes in function spaces (vol. 149). Lecture notes in statistics theory and applications. New York: Springer.Google Scholar
  8. Cobb, G. (1978). The problem of the nile: Conditional solution to a change-point problem. Biometrika, 65, 243–251.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cuevas, A. (2014). A partial overview of the theory of statistics with functional data. Journal of Statistical Planning and Inference, 147, 1–23.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Davis, R., Huang, D., Yao, Y.-C. (1995). Testing for a change in the parameter values and order of an autoregressive model. The Annals of Statistics, 23, 282–304.Google Scholar
  11. Ferraty, F. (2011). Recent advances in functional data analysis and related topics. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  12. Ferraty, F., Vieu, P. (2006). Nonparametric functional data analysis: Theory and practice. Chicago: Springer.Google Scholar
  13. Fotopoulos, S., Jandhyala, V. (2010). Exact asymptotic distribution of change point mle for change in the mean of gaussian sequences. Annals of Applied Statistics, 4, 1081–1104.Google Scholar
  14. Goia, A., Vieu, P. (2016). An introduction to recent advances in high/infinite dimensional statistics. Journal of Multivariate Analysis, 146, 1–6. Special Issue on Statistical models and methods for high or infinite dimensional spaces.Google Scholar
  15. Gombay, E., Horváth, L. (1994). An application of the maximum likelihood test to the change-point problem. Stochastic Processes and their Applications, 50, 1–17.Google Scholar
  16. Hadley Centre for Climate Prediction and Research (2007). Daily Mean, Minimum and Maximum Central England Temperature series. National Centre for Atmospheric Science (NCAS) British Atmospheric Data Centre. Accessed 12 Feb 2015.
  17. Hinkley, D. V. (1970). Inference about the change-point in a sequence of random variables. Biometrika, 57, 1–17.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hörmann, S., Kokoszka, P. (2010). Weakly dependent functional data. The Annals of Statistics, 38(3), 1845–1884.Google Scholar
  19. Horváth, L., Kokoszka, P. (2012). Inference for functional data with applications. Chicago: Springer.Google Scholar
  20. Horváth, L., Kokoszka, P., Steinebach, J. (1999). Testing for changes in multivariate dependent observations with applications to temperature changes. Journal of Multivariate Analysis, 68, 96–119.Google Scholar
  21. Horváth, L., Hušková, M., Rice, G. (2013). Test of independence for functional data. Journal of Multivariate Analysis, 117, 100–119.Google Scholar
  22. Hsing, T., Eubank, R. (2015). Theoretical foundations of functional data analysis, with an introduction to linear operators (1st ed.). Harvard: Wiley. Series in Probability and statistics.Google Scholar
  23. Inclán, C., Tiao, G. C. (1994). Use of cumulative sums of squares for retrospective detection of changes of variance. Journal of the American Statistical Association, 89(427), 913–923.Google Scholar
  24. Indritz, J. (1963). Methods in analysis. New York: The Macmillan Co.Google Scholar
  25. Jones, P., Parker, D., Osborn, T., Briffa, K. (2013). Global and hemispheric temperature anomalies land and marine instrumental records. In Trends: A compendium of data on global change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, U.S. Department of Energy, Oak Ridge,Tennessee.Google Scholar
  26. Kiefer, J. (1959). K-sample analogues of the kolmogorov-smirnov and cramér-v. mises tests. The Annals of Mathematical Statistics, 30, 420–447.Google Scholar
  27. Kirch, C., Muhsal, B., Ombao, H. (2015). Detection of changes in multivariate time series with application to eeg data. Journal of the American Statistical Association, 110(511), 1197–1216.Google Scholar
  28. Kokoszka, P., Leipus, R. (2000). Change-point estimation in arch models. Bernoulli, 6, 513–539.Google Scholar
  29. Mei, Y. (2006). Sequential change-point detection when unknown parameters are present in the pre-change distribution. The Annals of Statistics, 34, 92–122.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Ramsay, J. O., Silverman, B. (2005). Functional data analysis. New York: Springer.Google Scholar
  31. Rao, R. R. (1962). Relations between weak and uniform convergence of measures with applications. The Annals of Mathematical Statistics, 33, 659–680.Google Scholar
  32. Shao, X., Zhang, X. (2010). Testing for change point in time series. Journal of American Statistical Association, 105, 1228–1240.Google Scholar
  33. Zhang, J.-T. (2013). Analysis of variance for functional data. Harvard: CRC Press.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia
  2. 2.Department of Mathematics and StatisticsIndian Institute of Science Education and Research KolkataMohanpurIndia

Personalised recommendations