Abstract
In practice, the use of functional data is often preferable to that of large finitedimensional 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.
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References
Beente, G. and R. Fraiman (1989). Robust nonparametric regression estimation.Journal of Multivariate Analysis,29, 180–198.
Brown, B.M. (1983). Statistical uses of the spatial median.Journal of the Royal Statistical Society, B,45, 25–30.
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.
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.
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.
Fraiman, R. and J. Meloche (1999). Multivariate L-estimation.Test,8, 255–317.
Gordaliza, A. (1991). Best approximations to random variables based on trimming procedures.Journal of Approximation Theory,64, 162–180.
Härdle, W. and A.B. Tsybakov (1988). Robust nonparametric regression with simultaneous scale curve estimation.The Annals of Statistics,16, 120–135.
Liu, R. (1988). On a notion of simplicial depth.Proceedings of the National Academy of Sciences, U.S.A.,85, 1732–1734.
Liu, R. (1990). On a notion of data depth based on random simplices.The Annals of Statistics,18, 405–414.
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.
Mahalanobis, P.C. (1936). On the generalized distance in Statistics.Proceedings of the National Academy of India,12, 49–55.
Oja, H. (1983). Descriptive statistics for multivariate distributions.Statistics and Probability Letters,1, 327–332.
Pollard, D. (1984).Convergence of stochastic processes. Springer Verlag.
Ramsay, J.O. and B.W. Silverman (1997).Functional Data Analysis. Springer Verlag.
Singh, K. (1991). A notion of majority depth. Technical Report, Rutgers University, Department of Statistics.
Small, C.G. (1990). A survey of multidimensional medians.Intermational Statistical Review,58, 263–277.
Tukey, J.W. (1975). Mathematics and picturing data.Proceedings of the International Congress of Mathematics, Vancouver,2, 523–531.
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Fraiman, R., Muniz, G. Trimmed means for functional data. Test 10, 419–440 (2001). https://doi.org/10.1007/BF02595706
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DOI: https://doi.org/10.1007/BF02595706