Artículo
Abstract:
This paper deals with the maximum asymptotic bias of two classes of robust estimates of the dispersion matrix V of a p-dimensional random vector x, under a contamination model of the form P = (1 - ∊)P0 + δ∊(x0), where P is the distribution of x, P0 is a spherical distribution, and δ (x0) is a point mass at x0. Estimators VQ,α of the first class minimize the a quantile of x1V-1x among all symmetric positive-definite matrices V for some a ε (0,1). The “mimimum volume ellipsoid” estimator proposed by Rousseeuw belongs to this class with a = 0.5. These estimators have breakdown point min(α, 1 —α) for all p. The second class of estimators consist of the M-estimators, from which the seemingly most robust member was chosen; namely the Tyler estimate defined as the solution VT of Ex1VT-1x/x'x = VT. This estimator has breakdown point 1/p. The numerical results show that except for ε very close to 1/p, V.r has in general a smaller maximum bias than VQ,α; and that the maximum bias of the latter may be extremely large even for ε much smaller than its breakdown point. © 1990, Taylor & Francis Group, LLC. All rights reserved.
Registro:
| Documento: | Artículo |
| Título: | The maximum bias of robust covariances |
| Autor: | Yohai, V.J.; Maronna, R.A. |
| Filiación: | Departamento de Matematicas, Facultad de Ciencias Exactas, U.B.A., Ciudad Universitaria, Pabellon 1, Buenos Aires, Argentina, CONICET, Buenos Aires, Argentina Departamento de Matematicas, Facultad de Ciencias Exactas, La Plata. Argentina, U.N.L.P., c.c. No.172, C.I.C.P.B.A., La Plata, Argentina |
| Palabras clave: | breakdown point estimators; high; M-estimators; maximum bias; Robust covariance |
| Año: | 1990 |
| Volumen: | 19 |
| Número: | 10 |
| Página de inicio: | 3925 |
| Página de fin: | 3933 |
| DOI: | http://dx.doi.org/10.1080/03610929008830422 |
| Título revista: | Communications in Statistics - Theory and Methods |
| Título revista abreviado: | Commun Stat Theory Methods |
| ISSN: | 03610926 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03610926_v19_n10_p3925_Yohai |
Referencias:
- Donoho, D.L., Breakdown properties of multivariate location estimators (1982), Ph. D. qualifying paper Harvard University; Davies, P.L., Asymptotic behavior of S-estimates of multivariate location parameters and dispersion matrices (1987) Annals of Statististies, 15 (1269)
- Huber, P.J., Robust Statistics (1981), New York: Wiley; Maronna, R.A., Robust M-estimators of multivariate location and scatter (1976) Annals of Statistics., 4, pp. 51-67
- Martin, R.D., 0Yohai, V.J., Zamar, R., Minimax bias robust estimation of regression. Technical (1987), Report No. 112, University of Washington, Seattle, WA; Rousseeuw, P.J., Multivariate estimation with hight breakdown point. In (1986) Mathematical Statistics and Applications, pp. 283-297. , edited by W. Grossmann, G. Pflug, I. Vincze and W. Wertz, Reidel Publishing Company
- Stahel, W., Breakdown of covariance estimators (1981) Research Report No. 31, , Fachgruppe für Statistik, ETH, Zurich
- Tyler, D.E., A distribution free M-estimator of multivariate scatter (1987) Annals of Statistics., 15, pp. 234-251
Citas:
---------- APA ----------
Yohai, V.J. & Maronna, R.A. (1990) . The maximum bias of robust covariances. Communications in Statistics - Theory and Methods, 19(10), 3925-3933.http://dx.doi.org/10.1080/03610929008830422
---------- CHICAGO ----------
Yohai, V.J., Maronna, R.A. "The maximum bias of robust covariances" . Communications in Statistics - Theory and Methods 19, no. 10 (1990) : 3925-3933.http://dx.doi.org/10.1080/03610929008830422
---------- MLA ----------
Yohai, V.J., Maronna, R.A. "The maximum bias of robust covariances" . Communications in Statistics - Theory and Methods, vol. 19, no. 10, 1990, pp. 3925-3933.http://dx.doi.org/10.1080/03610929008830422
---------- VANCOUVER ----------
Yohai, V.J., Maronna, R.A. The maximum bias of robust covariances. Commun Stat Theory Methods. 1990;19(10):3925-3933.http://dx.doi.org/10.1080/03610929008830422
