×

zbMATH — the first resource for mathematics

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.

MSC:
62G07 Density estimation
62M09 Non-Markovian processes: estimation
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
Software:
fda (R)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Beente, G. and R. Fraiman (1989). Robust nonparametric regression estimation.Journal of Multivariate Analysis,29, 180–198. · Zbl 0688.62027 · doi:10.1016/0047-259X(89)90023-7
[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 · doi:10.1214/aos/1176348890
[6] Fraiman, R. and J. Meloche (1999). Multivariate L-estimation.Test,8, 255–317. · Zbl 0942.62062 · doi:10.1007/BF02595872
[7] Gordaliza, A. (1991). Best approximations to random variables based on trimming procedures.Journal of Approximation Theory,64, 162–180. · Zbl 0745.41030 · doi:10.1016/0021-9045(91)90072-I
[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 · doi:10.1214/aos/1176350694
[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 · doi:10.1073/pnas.85.6.1732
[10] Liu, R. (1990). On a notion of data depth based on random simplices.The Annals of Statistics,18, 405–414. · Zbl 0701.62063 · doi:10.1214/aos/1176347507
[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 · doi:10.2307/2290720
[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 · doi:10.1016/0167-7152(83)90054-8
[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. · doi:10.2307/1403809
[18] Tukey, J.W. (1975). Mathematics and picturing data.Proceedings of the International Congress of Mathematics, Vancouver,2, 523–531. · Zbl 0347.62002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.