Trimmed means for functional data. (English) Zbl 1016.62026

Summary: In practice, the use of functional data is often preferable to that of large finite-dimensional vectors obtained by discrete approximations of functions. In this paper a new concept of data depth is introduced for functional data. The aim is to measure the centrality of a given curve within a group of curves. This concept is used to define ranks and trimmed means for functional data. Some theoretical and practical aspects are discussed and a simulation study is given. The results show a good performance of our method, in terms of efficiency and robustness, when compared with the mean. Finally, a real-data example based on the Nasdaq 100 index is discussed.


62G07 Density estimation
62M09 Non-Markovian processes: estimation
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation


fda (R)
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[1] Beente, G. and R. Fraiman (1989). Robust nonparametric regression estimation.Journal of Multivariate Analysis,29, 180–198. · Zbl 0688.62027
[2] Brown, B.M. (1983). Statistical uses of the spatial median.Journal of the Royal Statistical Society, B,45, 25–30. · Zbl 0508.62046
[3] Brown, B.M., and T.P. Hettmansperger (1987). Affine invariant rank methods in the bivariate location model.Journal of the Royal Statistical Society, B,49, 301–310. · Zbl 0653.62039
[4] Cuesta-Albertos, J.A., A. Gordaliza and C. Matrán (1998). Trimmed k-means: An attempt to robustify quantizers.The Annals of Statistics,25, 553–576. · Zbl 0878.62045
[5] Donoho, D.L. and M. Gasko (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness.The Annals of Statistics,20, 1803–1827. · Zbl 0776.62031
[6] Fraiman, R. and J. Meloche (1999). Multivariate L-estimation.Test,8, 255–317. · Zbl 0942.62062
[7] Gordaliza, A. (1991). Best approximations to random variables based on trimming procedures.Journal of Approximation Theory,64, 162–180. · Zbl 0745.41030
[8] Härdle, W. and A.B. Tsybakov (1988). Robust nonparametric regression with simultaneous scale curve estimation.The Annals of Statistics,16, 120–135. · Zbl 0668.62025
[9] Liu, R. (1988). On a notion of simplicial depth.Proceedings of the National Academy of Sciences, U.S.A.,85, 1732–1734. · Zbl 0635.62039
[10] Liu, R. (1990). On a notion of data depth based on random simplices.The Annals of Statistics,18, 405–414. · Zbl 0701.62063
[11] Liu, R. and K. Singh (1993). A quality index based on data depth and multivariate rank tests.Journal of the American Statistical Association,421, 252–260. · Zbl 0772.62031
[12] Mahalanobis, P.C. (1936). On the generalized distance in Statistics.Proceedings of the National Academy of India,12, 49–55. · Zbl 0015.03302
[13] Oja, H. (1983). Descriptive statistics for multivariate distributions.Statistics and Probability Letters,1, 327–332. · Zbl 0517.62051
[14] Pollard, D. (1984).Convergence of stochastic processes. Springer Verlag. · Zbl 0544.60045
[15] Ramsay, J.O. and B.W. Silverman (1997).Functional Data Analysis. Springer Verlag. · Zbl 0882.62002
[16] Singh, K. (1991). A notion of majority depth. Technical Report, Rutgers University, Department of Statistics.
[17] Small, C.G. (1990). A survey of multidimensional medians.Intermational Statistical Review,58, 263–277.
[18] Tukey, J.W. (1975). Mathematics and picturing data.Proceedings of the International Congress of Mathematics, Vancouver,2, 523–531. · Zbl 0347.62002
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