• Bernoulli
  • Volume 24, Number 3 (2018), 2328-2357.

M-estimators of location for functional data

Beatriz Sinova, Gil González-Rodríguez, and Stefan Van Aelst

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


M-estimators of location are widely used robust estimators of the center of univariate or multivariate real-valued data. This paper aims to study M-estimates of location in the framework of functional data analysis. To this end, recent developments for robust nonparametric density estimation by means of M-estimators are considered. These results can also be applied in the context of functional data analysis and allow to state conditions for the existence and uniqueness of location M-estimates in this setting. Properties of these functional M-estimators are investigated. In particular, their consistency is shown and robustness is studied by means of their breakdown point and their influence function. The finite-sample performance of the M-estimators is explored by simulation. The M-estimators are also empirically compared to trimmed means for functional data.

Article information

Bernoulli, Volume 24, Number 3 (2018), 2328-2357.

Received: February 2016
Revised: October 2016
First available in Project Euclid: 2 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

functional data functional data metric Hampel loss Huber loss M-estimates of location statistical robustness trimmed means Tukey loss


Sinova, Beatriz; González-Rodríguez, Gil; Van Aelst, Stefan. M-estimators of location for functional data. Bernoulli 24 (2018), no. 3, 2328--2357. doi:10.3150/17-BEJ929.

Export citation


  • [1] Beaton, A.E. and Tukey, J.W. (1974). The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics 16 147–185.
  • [2] Brown, L.D. and Purves, R. (1973). Measurable selections of extrema. Ann. Statist. 1 902–912.
  • [3] Chichignoud, M. and Lederer, J. (2014). A robust, adaptive M-estimator for pointwise estimation in heteroscedastic regression. Bernoulli 20 1560–1599.
  • [4] Christmann, A. and Steinwart, I. (2007). Consistency and robustness of kernel-based regression in convex risk minimization. Bernoulli 13 799–819.
  • [5] Ciarleglio, A. and Ogden, R.T. (2016). Wavelet-based scalar-on-function finite mixture regression models. Comput. Statist. Data Anal. 93 86–96.
  • [6] Claeskens, G., Hubert, M., Slaets, L. and Vakili, K. (2014). Multivariate functional halfspace depth. J. Amer. Statist. Assoc. 109 411–423.
  • [7] Colubi, A. and González-Rodríguez, G. (2015). Fuzziness in data analysis: Towards accuracy and robustness. Fuzzy Sets and Systems 281 260–271.
  • [8] Cuesta-Albertos, J.A. and Fraiman, R. (2007). Impartial trimmed $k$-means for functional data. Comput. Statist. Data Anal. 51 4864–4877.
  • [9] Cuesta-Albertos, J.A. and Nieto-Reyes, A. (2008). The random tukey depth. Comput. Statist. Data Anal. 52 4979–4988.
  • [10] Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Comput. Statist. 22 481–496.
  • [11] Cuevas, A. and Fraiman, R. (2009). On depth measures and dual statistics. A methodology for dealing with general data. J. Multivariate Anal. 100 753–766.
  • [12] D’Urso, P. and De Giovanni, L. (2014). Robust clustering of imprecise data. Chemom. Intell. Lab. Syst. 136 58–80.
  • [13] D’Urso, P., De Giovanni, L. and Massari, R. (2015). Trimmed fuzzy clustering for interval-valued data. Adv. Data Anal. Classif. 9 21–40.
  • [14] D’Urso, P., Massari, R. and Santoro, A. (2011). Robust fuzzy regression analysis. Inform. Sci. 181 4154–4174.
  • [15] Domingues, M.A.O., De Souza, R.M.C.R. and Cysneiros, F.J.A. (2010). A robust method for linear regression of symbolic interval data. Pattern Recogn. Lett. 31 1991–1996.
  • [16] Donoho, D. and Huber, P.J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P.J. Bickel, K. Doksum and J.L. Hodges, eds.). Wadsworth Statist./Probab. Ser. 157–184. Belmont, CA: Wadsworth.
  • [17] Ferraty, F. and Vieu, P. (2006). Non Parametric Functional Data Analysis. Theory and Practice. Berlin: Springer.
  • [18] Filzmoser, P. and Todorov, V. (2013). Robust tools for the imperfect world. Inform. Sci. 245 4–20.
  • [19] Fraiman, R. and Muniz, G. (2001). Trimmed means for functional data. TEST 10 419–440.
  • [20] Fraiman, R. and Svarc, M. (2013). Resistant estimates for high dimensional and functional data based on random projections. Comput. Statist. Data Anal. 58 326–338.
  • [21] Frappart, F. (2003). Catalogue des formes d’onde de l’altimètre Topex/Poséidon sur le bassin amazonien. Toulouse, France: Tec. Rep., CNES.
  • [22] Fritz, H., García-Escudero, L.A. and Mayo-Iscar, A. (2013). Robust constrained fuzzy clustering. Inform. Sci. 245 38–52.
  • [23] González, J., Vujačić, I. and Wit, E. (2014). Reproducing kernel Hilbert space based estimation of systems of ordinary differential equations. Pattern Recogn. Lett. 45 26–32.
  • [24] González-Rodríguez, G., Colubi, A. and Gil, M.A. (2012). Fuzzy data treated as functional data. A one-way ANOVA test approach. Comput. Statist. Data Anal. 56 943–955.
  • [25] Hampel, F.R. (1971). A general qualitative definition of robustness. Ann. Math. Stat. 42 1887–1896.
  • [26] Hampel, F.R. (1974). The influence curve and its role in robust estimation. J. Amer. Statist. Assoc. 69 383–393.
  • [27] Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J. and Stahel, W.A. (1986). Robust Statistics: The Approach Based on Influence Functions. New York: Wiley.
  • [28] Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Chichester: Wiley.
  • [29] Hu, Q., An, S., Yu, X. and Yu, D. (2011). Robust fuzzy rough classifiers. Fuzzy Sets and Systems 183 26–43.
  • [30] Huber, P.J. (1964). Robust estimation of a location parameter. Ann. Math. Stat. 35 73–101.
  • [31] Huber, P.J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. I: Statistics 221–233. Berkeley, CA: Univ. California Press.
  • [32] Huber, P.J. (1981). Robust Statistics. New York: Wiley.
  • [33] Hubert, M., Rousseeuw, P. and Segaert, P. (2015). Multivariate functional outlier detection. Stat. Methods Appl. 24 177–202.
  • [34] Jacques, J. and Preda, C. (2014). Model-based clustering for multivariate functional data. Comput. Statist. Data Anal. 71 92–106.
  • [35] Jerrard, R.L. and Sternberg, P. (2009). Critical points via $\Gamma$-convergence: General theory and applications. J. Eur. Math. Soc. (JEMS) 11 705–753.
  • [36] Kim, J. and Scott, C.D. (2012). Robust kernel density estimation. J. Mach. Learn. Res. 13 2529–2565.
  • [37] Kim, J.S. (2011). Kernel methods for classification with irregularly sampled and contaminated data. PhD thesis, University of Michigan. Available at
  • [38] Kim, J.S. and Scott, C.D. (2011). On the robustness of kernel density M-estimators. In Proc. 28th Int. Conf. Mach. Lear. 697–704. Washington: Bellevue.
  • [39] López-Pintado, S. and Romo, J. (2009). On the concept of depth for functional data. J. Amer. Statist. Assoc. 104 718–734.
  • [40] López-Pintado, S. and Romo, J. (2011). A half-region depth for functional data. Comput. Statist. Data Anal. 55 1679–1695.
  • [41] Lopuhaä, H.P. and Rousseeuw, P.J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Statist. 19 229–248.
  • [42] Ma, J., Zhao, J., Tian, J., Yuille, A.L. and Tu, Z. (2014). Robust point matching via vector field consensus. IEEE Trans. Image Process. 23 1706–1721.
  • [43] Maronna, R.A., Martin, R.D. and Yohai, V.J. (2006). Robust Statistics: Theory and Methods. Chichester: Wiley.
  • [44] Minsker, S. (2015). Geometric median and robust estimation in Banach spaces. Bernoulli 21 2308–2335.
  • [45] Ordóñez-Cabrera, M., Rosalsky, A. and Volodin, A. (2012). Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables. TEST 21 369–385.
  • [46] Paulauskas, V. and Račkauskas, A. (1989). Approximation Theory in the Central Limit Theorem. Mathematics and Its Applications (Soviet Series) 32. Dordrecht: Kluwer Academic.
  • [47] Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [48] Rockafellar, R.T. and Wets, R.J.B. (1998). Variational Analysis. Heidelberg: Springer.
  • [49] Rosen, B.R., Buckner, R.L. and Dale, A.M. (1998). Event-related functional MRI: Past, present, and future. Proc. Natl. Acad. Sci. USA 95 773–780.
  • [50] Rousseeuw, P.J. and Van Driessen, K. (2006). Computing LTS regression for large data sets. Data Min. Knowl. Discov. 12 29–45.
  • [51] Sinova, B., Gil, M.Á., Colubi, A. and Van Aelst, S. (2012). The median of a random fuzzy number. The 1-norm distance approach. Fuzzy Sets and Systems 200 99–115.
  • [52] Sinova, B., Gil, M.Á., López, M.T. and Van Aelst, S. (2014). A parameterized $L^{2}$ metric between fuzzy numbers and its parameter interpretation. Fuzzy Sets and Systems 245 101–115.
  • [53] Sinova, B., Pérez-Fernández and Montenegro M, S. (2015). The wabl/ldev/rdev median of a random fuzzy number and statistical properties. In Strengthening Links Between Data Analysis and Soft Computing (P. Grzegorzewski, M. Gagolewski, O. Hryniewicz and M.A. Gil, eds.) 143–150. Heidelberg: Springer.
  • [54] Sreenivasa, M., Souères, P. and Laumond, J.P. (2012). Walking to grasp: Modeling of human movements as invariants and an application to humanoid robotics. IEEE Trans. Syst. Man Cybern. Syst. 42 880–893.
  • [55] Steinwart, I. and Christmann, A. (2008). Support Vector Machines. New York: Springer.
  • [56] Sziláagyi, L. (2013). Robust spherical shell clustering using fuzzy-possibilistic product partition. Int. J. Intel. Syst. 28 524–539.
  • [57] Vakhania, N.N., Tarieladze, V.I. and Chobanyan, S.A. (1987). Probability Distributions on Banach Spaces. Dordrecht: D. Reidel Publishing Co.
  • [58] Vandermeulen, R.A. and Scott, C.D. (2013). Consistency of robust kernel density estimators. JMLR: Work. Conf. Proc. 30 1–24.
  • [59] Winkler, R., Klawonn, F. and Kruse, R. (2011). Fuzzy clustering with polynomial fuzzifier function in connection with $M$-estimators. Appl. Comput. Math. 10 146–163.
  • [60] Wood, A.T.A. and Chan, G. (1994). Simulation of stationary Gaussian processes in $[0,1]^{d}$. J. Comput. Graph. Statist. 3 409–432.
  • [61] Zhang, Z. and Zhang, Y. (2014). Variable kernel density estimation based robust regression and its applications. Neurocomputing 134 30–37.