Abstract
The paper is concerned with the extreme behavior of projections of time series of functions onto data-driven basis systems, for example, on the estimated functional principal components. The coefficients of these projections, called the scores, encode the shapes of the curves. Within the framework of functional data analysis, the extreme shapes are those corresponding to multivariate extremes of the scores. The scores are not directly observable, and must be computed from the data. Even for iid Gaussian functions, they form a triangular array of dependent non–Gaussian vectors. Thus, even though the extreme behavior of the population scores of Gaussian functions follows from well–known results, it is not clear what the extreme behavior of their approximations computed from the data is. We clarify these issues for Gaussian functions and for more general functional time series whose projections are in the Gumbel domain of attraction.
Similar content being viewed by others
References
Aue, A., Hörmann, S., Horváth, L., Reimherr, M.: Break detection in the covariance structure of multivariate time series models. Ann. Stat. 37, 4046–4087 (2009)
Barbe, P.h., McCormick, W.P.: Second-order expansion for the maximum of some stationary gaussian sequences. Stochastic Processes and their Applications 110, 315–342 (2004)
Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of extremes: theory and applications. Wiley, New York (2006)
Dauxois, J., Pousse, A., Romain, Y.: Asymptotic theory for principal component analysis of a vector random function. J. Multivar. Anal. 12, 136–154 (1982)
Davis, R.A., Resnick, S.I.: Tail estimates motivated by extreme value theory. Ann. Stat. 12, 1467–1487 (1984)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling extremal events for insurance and finance. Springer, Berlin (1997)
Gumbel, E.J.: Statistics of extremes. Courier Corporation, North Chelmsford (2012)
de Haan, L., Ferreira, A.: Extreme value theory: an introduction. Springer, Berlin (2006)
Hörmann, S., Kidziński, L., Hallin, M.: Dynamic functional principal components. J. R. Stat. Soc. (B) 77, 319–348 (2015)
Hörmann, S., Kokoszka, P.: Weakly dependent functional data. Ann. Stat. 38, 1845–1884 (2010)
Horváth, L., Kokoszka, P.: Inference for functional data with applications. Springer, Berlin (2012)
Horváth, L., Kokoszka, P., Reeder, R.: Estimation of the mean of functional time series and a two sample problem. J. R. Stat. Soc. (B) 75, 103–122 (2013)
Hsing, T., Eubank, R.: Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley, New York (2015)
Hsing, T., Hüsler, J., Reiss, R.-D.: The extremes of a triangular array of normal random variables. Ann. Appl. Probab. 6, 671–686 (1996)
Kokoszka, P., Reimherr, M.: Asymptotic normality of the principal components of functional time series. Stoch. Process. Appl. 123, 1546–1562 (2013)
Kokoszka, P., Reimherr, M.: Introduction to functional data analysis. CRC Press, Boca Raton (2017)
Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer, Boca Raton (1983)
Ramsay, J.O., Silverman, B.W.: Functional data analysis. Springer, Berlin (2005)
Resnick, S.I.: Extreme values, regular variation, and point processes. Springer, Berlin (1987)
Rootzén, H.: The rate of convergence of extremes of stationary normal sequences. Adv. Appl. Probab. 15, 54–80 (1983)
Shao, X., Wu, W.B.: Asymptotic spectral theory for nonlinear time series. Ann. Stat. 35, 1773–1801 (2007)
Vakhaniia, N.N., Tarieladze, V.I., Chobanian, S.A.: Probability distributions on banach spaces. Springer, Berlin (1987)
Wu, W.B.: Strong invariance principles for dependent random variables. Ann. Probab. 35, 2294–2320 (2007)
Zhang, X.: White noise testing and model diagnostic checking for functional time series. J. Econ. 194, 76–95 (2016)
Acknowledgments
This research was partially supported by NSF grant DMS 1462067 “FRG: Collaborative Research: Extreme Value Theory for Spatially Indexed Functional Data”. We thank Professor Haonan Wang for useful advice on elements of the proof in the Gaussian case, and two referees for valuable advice on both presentation and substance.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kokoszka, P., Xiong, Q. Extremes of projections of functional time series on data–driven basis systems. Extremes 21, 177–204 (2018). https://doi.org/10.1007/s10687-017-0302-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-017-0302-8