Abstract
In Ruiz-Gazen (Comput Stat Data Anal 21:149–162, 1996), a simple B-robust estimator was introduced. Its definition is explicit and takes into account the empirical covariance matrix together with a one-step M-estimator. In the present paper, we derive the asymptotics and some robustness properties of this estimator. We compare its performance to the usual M- and S-estimators by means of a Monte-Carlo study. We also illustrate its use for cluster detection using Invariant Coordinate Selection on a small example.
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References
Bedall, F.K., Zimmerman, H.: Algorithm AS 143, the mediancenter. App. Stat. 28, 325–328 (1979)
Béguin, C., Hulliger, B.: Multivariate outlier detection in incomplete survey data: the epidemic algorithm and transformed rank correlations. J. R. Stat. Soc. Ser. A (Stat. Soc.) 167(2), 275–294 (2004)
Cator, E.A., Lopuhaä, H.P.: Central limit theorem and influence function for the MCD estimators at general multivariate distributions. Bernoulli 18(2), 520–551 (2012)
Caussinus, H., Hakam, S., Ruiz-Gazen, A.: Projections révélatrices contrôlées: groupements et structures diverses. Rev. Stat. Appl. 51(1), 37–58 (2003)
Critchley, F.: Influence in principal components analysis. Biometrika 72, 627–636 (1985)
Croux, C., Haesbroeck, G.: Principal component analysis based on robust estimators of the covariance or correlation matrix : influence functions and efficiencies. Biometrika 87(3), 603–618 (2000)
Dauxois, J., Pousse, A., Romain, Y.: Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference. J. Multivar. Anal. 12, 136–154 (1982)
Davies, P.L.: Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices. Ann. Stat. 15(3), 1269–1292 (1987)
Davies, L.: The asymptotics of Rousseeuw’s minimum volume ellipsoid estimator. Ann. Stat. 20, 1828–1843 (1992)
Devlin, S.J., Gnanadesikan, R., Kettenring, J.R.: Robust estimation of dispersion matrices and principal components. J. Am. Stat. Assoc. 76, 354–362 (1981)
Donoho, D.L.: Breakdown properties of multivariate location estimators. Ph.D. Qualifying Paper. Department of Statistics, Harvard University (1982)
Dossou-Gbete, S., Pousse, A.: Asymptotic study of eigenelements of a sequence of random selfadjoint operators. Statistics 3, 479–491 (1991)
Dümbgen, L., Pauly, M., Schweizer, T.: A survey of M-functionals of multivariate location and scatter. arXiv preprint arXiv:1312.5594 (2013a)
Dümbgen, L., Nordhausen, K., Schuhmacher, H.: New algorithms for M-estimation of multivariate location and scatter. arXiv preprint arXiv:1312.6489 (2013b)
Fekri, M., Fine, J.: Matrice aléatoire dont l’espérance est de rang réduit. Propriétés asymptotiques des estimateurs moindres carrés et choix de métriques. Pub. Inst. Stat. Univ. Paris XXXIX(1), 67–88 (1995)
Fekri, M., Ruiz-Gazen, A.: Propriétés asymptotiques et fonction d’influence d’un estimateur simple et robuste de matrice de dispersion. C. R. Acad. Sci. Paris t. 330 série I, 565–568 (2000)
Fischer, D., Möttönen, J., Nordhausen, K., Vogel, H.: OjaNP: multivariate methods based on the Oja median and related concepts. R package vesion 0.9-6. http://cran.r-project.org/web/packages/OjaNP/index.html (2013)
Gervini, D.: The influence function of the Stahel-Donoho estimator of multivariate location and scatter. Stat. Probab. Lett. 60(4), 425–435 (2002)
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics. Wiley, New York (1986)
Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)
Le Cam, L.: On some asymptotic properties of maximum likelihood estimates and related Bayes estimates. Univ. Calif. Public Stat. 1, 277–330 (1953)
Lopuhaä, H.P.: Asymptotics of reweighted estimators of multivariate location and scatter. Ann. Stat. 27(5), 1638–1665 (1999)
Lubischew, A.A.: On the use of discriminant functions in taxonomy. Biometrics 18(4), 455–477 (1962)
Ma, Y., Genton, M.G.: Highly robust estimation of dispersion matrices. J. Multiv. Anal. 78(1), 11–36 (2001)
Maronna, R.A.: Robust M-estimators of multivariate location and scatter. Ann. Stat. 4, 51–67 (1976)
Maronna, R.A., Stahel, W.A., Yohai, V.J.: Bias-robust estimators of multivariate scatter based on projections. J. Multiv. Anal. 42(1), 141–161 (1992)
Maronna, R.A., Yohai, V.J.: Robust estimation of multivariate location and scatter. In: Kotz, S., Read, C., Banks, D. (eds.) Encyclopedia of Statistical Sciences, Update vol. 2, pp. 589–596. Wiley, New York (1988)
Maronna, R.A., Yohai, V.J.: The behavior of the Stahel-Donoho robust multivariate estimator. J. Am. Stat. Assoc. 90, 330–341 (1995)
Maronna, R.A., Zamar, R.H.: Robust estimates of location and dispersion for high-dimensional datasets. Technometrics 44(4), 307–317 (2002)
Meshalkin, L.D.: Approximation of multidimensional densities by normal distributions. In: Proceedings of 7th International Biometric Conference, Hannover (1970)
Oja, H.: Descriptive statistics for multivariate distributions. Stat. Probab. Lett. 1(6), 327–332 (1983)
Ollila, E., Croux, C., Oja, H.: Influence function and asymptotic efficiency of the affine equivariant rank covariance matrix. DTEW Research Report 0210, 1–19 (2002)
Ollila, E., Oja, H., Croux, C.: The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies. J. Multiv. Anal. 87(2), 328–355 (2003)
Peña, D., Prieto, F.J.: Cluster identification using projections. J. Am. Stat. Assoc. 96(456), 14433–1445 (2001)
Rousseeuw, P.J.: Multivariate estimation with high breakdown point. In: Grossmann, W., G. Pflug, G., I. Vincze, I., W. Wertz, W. (eds.) Mathematical Statistics and Applications, pp. 283–297. Reidel, Dordrecht (1985)
Rousseeuw, P.J., Croux, C.: The bias of k-step M-estimators. Stat. Probab. Lett. 20(5), 411–420 (1994)
Rousseeuw, P.J., Leroy, A.M.: Robust Regression and Outlier Detection. Wiley, New York (1987)
Ruiz-Gazen, A.: A very simple robust estimator of a dispersion matrix. Comput. Stat. Data Anal. 21, 149–162 (1996)
Stahel, W.A.: Breakdown of covariance estimators. Research Report 31. Fachgruppe für Statistik. ETH, Zürich (1981)
Tanaka, Y.: Sensitivity analysis in PCA: influence on the subspace spanned by principal components. Commun. Stat. Theory Methods 17, 3157–3175 (1988)
Tyler, D.E., Critchley, F., Dümbgen, L., Oja, H.: Invariant co-ordinate selection. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 71(3), 549–592 (2009)
Visuri, S., Koivunen, V., Oja, H.: Sign and rank covariance matrices. J. Stat. Plann. Inf. 91(2), 557–575 (2000)
Zuo, Y., Cui, H., He, X.: On the Stahel–Donoho estimators and depth-weighted means for multivariate data. Ann. Stat. 32(1), 167–188 (2004)
Zuo, Y., Lai, S.: Exact computation of bivariate projection depth and the Stahel–Donoho estimator. Comput. Stat. Data Anal. 55(3), 1173–1179 (2011)
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Fekri, M., Ruiz-Gazen, A. (2015). A B-Robust Non-Iterative Scatter Matrix Estimator: Asymptotics and Application to Cluster Detection Using Invariant Coordinate Selection. In: Nordhausen, K., Taskinen, S. (eds) Modern Nonparametric, Robust and Multivariate Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-22404-6_22
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DOI: https://doi.org/10.1007/978-3-319-22404-6_22
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