Abstract
Five notions of data depth are considered. They are mostly designed for functional data but they can be also adapted to the standard multivariate case. The performance of these depth notions, when used as auxiliary tools in estimation and classification, is checked through a Monte Carlo study.
Similar content being viewed by others
References
Biau G, Bunea F, Wegkamp M (2005) Functional classification in Hilbert spaces. IEEE Trans Inf Theory 51:2163–2172
Cérou F, Guyader A (2005) Nearest neighbor classification in infinite dimension. Preprint
Cuesta-Albertos JA, Fraiman R, Ransford T (2006a) A sharp form of the Cramer–Wold theorem. J Theor Probab (in press)
Cuesta-Albertos JA, Fraiman R, Ransford T (2006b) Random projections and goodness-of-fit tests in infinite-dimensional spaces. Boletim da Sociedade Brasileira de Matematica 37:1–25
Cuevas A, Febrero M, Fraiman R (2006) On the use of the bootstrap for estimating functions with functional data. Comput Stat Data Anal 51:1063–1074
Devroye L, Györfi L, Lugosi G (1996) A probabilistic theory of pattern recognition. Springer, Heidelberg
Donoho DL (1982) Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Dept. Statist., Harvard University
Ferraty F, Vieu P (2003) Curves discrimination: a nonparametric functional approach. Comput Stat Data Anal 44:161–173
Ferraty F, Vieu P (2006) Nonparametric modelling for functional data. Springer, Heidelberg
Fishman GS (1996) Monte Carlo: concepts, algorithms and applications. Springer, Heidelberg
Fraiman R, Meloche J (1999) Multivariate L-estimation. Test 8:255–317
Fraiman R, Muniz G (2001) Trimmed means for functional data. Test 10:419–440
Ghosh AK, Chaudhuri P (2005) On data depth and distribution-free discriminant analysis using separating surfaces. Bernoulli 11:1–27
Hastie T, Buja A, Tibshirani R (1995) Penalized discriminant analysis. Ann Stat 23:73–102
Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Data mining, inference, and prediction. Springer, New York
James GM, Hastie TJ (2001) Functional linear discriminant analysis for irregularly sampled curves. J R Stat Soc B 63:533–550
Liu RY (1990) On a notion of data depth based on random simplices. Ann Stat 18:405–414
Liu RY, Parelius JM, Singh K (1999) Multivariate analysis by data depth: descriptive statistics, graphics and inference. With discussion and a reply by the authors. Ann Stat 27:783–858
Ramsay JO, Silverman BW (2002) Applied functional data analysis. Methods and case studies. Springer, New York
Ramsay JO, Silverman BW (2005a) Applied functional data analysis. Springer, Heidelberg
Ramsay JO, Silverman BW (2005b) Functional data analysis, 2nd edn. Springer, Heidelberg
Rousseeuw PJ, Hubert M (1999) Regression depth (with discussion). J Am Stat Assoc 94:388–433
Stone CJ (1977) Consistent nonparametric regression. With discussion and a reply by the author. Ann Stat 5:595–645
Tukey JW (1975) Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, pp 523–531. Canad. Math. Congress, Montreal
Zuo Y (2003) Projection based depth functions and associated medians. Ann Stat 31:1460–1490
Zuo Y, Serfling R (2000) General notions of statistical depth function. Ann Stat 28:461–482
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by Spanish grants MTM2004-00098 (A. Cuevas and R. Fraiman) and MTM2005-00820 (M. Febrero).
Rights and permissions
About this article
Cite this article
Cuevas, A., Febrero, M. & Fraiman, R. Robust estimation and classification for functional data via projection-based depth notions. Computational Statistics 22, 481–496 (2007). https://doi.org/10.1007/s00180-007-0053-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-007-0053-0