Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

The common principal components model for several groups of multivariate observations is a useful parsimonious model for the scatter structure which assumes equal principal axes but different variances along those axes for each group. Due to the lack of resistance of the classical maximum likelihood estimators for the parameters of this model, several robust estimators have been proposed in the literature: plug-in estimators and projection-pursuit (PP) type estimators. In this paper, we show that it is possible to improve the low efficiency of the projection-pursuit estimators by applying a reweighting step. More precisely, we consider plug-in estimators obtained by plugging a reweighted estimator of the scatter matrices into the maximum likelihood equations defining the principal axes. The weights considered penalize observations with large values of the influence measures defined by Boente et al. (2002). The new estimators are studied in terms of theoretical properties (influence functions and asymptotic variances) and are compared with other existing estimators in a simulation study. © 2007 Springer-Verlag.

Registro:

Documento: Artículo
Título:Estimators for the common principal components model based on reweighting: Influence functions and Monte Carlo study
Autor:Boente, G.; Pires, A.M.; Rodrigues, I.M.
Filiación:CONICET, Departamento de Matemática, Ciudad Universitaria, Pabellón 2, Buenos Aires C1428EHA, Argentina
Departamento de Matemática, CEMAT, Technical University of Lisbon (TULisbon), Avda. Rovisco Pais, 1, 1049-001 Lisboa, Portugal
Palabras clave:Common principal components; Outlier detection; Projection-Pursuit; Reweighted estimators; Robust estimation
Año:2008
Volumen:67
Número:2
Página de inicio:189
Página de fin:218
DOI: http://dx.doi.org/10.1007/s00184-007-0129-4
Título revista:Metrika
Título revista abreviado:Metrika
ISSN:00261335
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00261335_v67_n2_p189_Boente

Referencias:

  • Adrover, J., Minimax bias-robust estimation of the dispersion matrix of a multivariate distribution (1998) Ann Stat, 26, pp. 2301-2320
  • Boente, G., Orellana, L., Fernholz, L., Morgenthaler, S., Stahel, W., A robust approach to common principal components (2001) Statistics in Genetics and in the Environmental Sciences, pp. 117-147. , Birkhauser Basel
  • Boente, G., Pires, A.M., Rodrigues, I.M., Influence functions and outlier detection under the common principal components model: A robust approach (2002) Biometrika, 89, pp. 861-875
  • Boente, G., Critchley, F., Orellana, L., (2004) Influence Functions for Robust Estimators under Proportional Scatter Matrices, , http://www.ic.fcen.uba.ar/preprints/boecriore.pdf
  • Boente, G., Pires, A.M., Rodrigues, I.M., (2005) Reweighted Estimators for the Common Principal Components Model: Influence Functions and Monte Carlo Study, , http://www.ic.fcen.uba.ar/preprints/boentepiresrodrigues.pdf, Publicaciones Previas del Instituto de Cálculo
  • Boente, G., Pires, A.M., Rodrigues, I.M., General projection-pursuit estimators for the common principal components model: Influence functions and Monte Carlo study (2006) J Multivariate Anal, 97, pp. 124-147
  • Critchley, F., Influence in principal components analysis (1985) Biometrika, 72, pp. 627-636
  • Croux, C., Haesbroeck, G., Influence function and efficiency of the minimum covariance determinant scatter matrix estimator (1999) J Multivariate Anal, 71, pp. 161-190
  • Croux, C., Haesbroeck, G., Principal component analysis based on robust estimators of the covariance or correlation matrix: Influence functions and efficiencies (2000) Biometrika, 87, pp. 603-618
  • Croux, C., Ruiz-Gazen, A., High breakdown estimators for principal components: The projection-pursuit approach revisited (2005) J Multivariate Anal, 95, pp. 206-226
  • Davies, P.L., Asymptotic behavior of S-estimates of multivariate location parameters and dispersion matrices (1987) Ann Stat, 15, pp. 1269-1292
  • Donoho, D.L., (1982) Breakdown Properties of Multivariate Location Estimators, Qualifying Paper, , Harvard University Boston
  • Filzmoser, P., Aivazian, S., Filzmoser, P., Kharin, Y., A multivariate outlier detection method (2004) Proceedings of the Seventh International Conference on Computer Data Analysis and Modelling, Vol 1, pp. 18-22. , Belarusian State University Minsk
  • Filzmoser, P., Reimann, C., Garrett, R.G., Multivariate outlier detection in exploration geochemistry (2005) Comput Geosci, 31, pp. 579-587
  • Flury, B.K., Common principal components in k groups (1984) J Am Stat Assoc, 79, pp. 892-898
  • Flury, B.K., (1988) Common Principal Components and Related Multivariate Models, , Wiley New York
  • Huber, P., Guta, S.S., Moore, D.S., Robust covariances (1977) Statistical Decision Theory and Related Topics, pp. 165-191. , Academic New York
  • Jaupi, L., Saporta, G., Morgenthaler, S., Ronchetti, E., Stahel, W., Using the influence function in robust principal components analysis (1993) New Directions in Statistical Data Analysis and Robustness, pp. 147-156. , Birkhauser Basel
  • Kent, J., Tyler, D., Constrained M-estimation for multivariate location and scatter (1996) Ann Stat, 24, pp. 1346-1370
  • Li, G., Chen, Z., Projection-pursuit approach to robust dispersion matrices and principal components: Primary theory and Monte Carlo (1985) J Am Stat Assoc, 80, pp. 759-766
  • Lopuhaä, H.P., Asymptotics of reweighted estimators of multivariate location and scatter (1999) Ann Stat, 27, pp. 1-28
  • Maronna, R., Robust M-estimates of multivariate location and scatter (1976) Ann Stat, 4, pp. 51-67
  • Maronna, R., Yohai, V., Kotz, S., Nl, J., Robust estimation of multivariate locaton and scatter (1998) Encyclopedia of Statistical Sciences, Update Vol 2, pp. 589-596. , Wiley New York
  • Maronna, R., Martin, D., Yohai, V., (2006) Robust Statistics: Theory and Methods, , Wiley New York
  • Pires, A.M., Branco, J., Partial influence functions (2002) J Multivariate Anal, 83, pp. 451-468
  • Pison, G., Rousseeuw, P.J., Filzmoser, P., Croux, C., Bethlehem, J., Van Der Heijden, P., A robust version of principal factor analysis (2000) Compstat: Proceedings in Computational Statistics, pp. 385-390. , Physica-Verlag Heidelberg
  • Pison, G., Rousseeuw, P.J., Filzmoser, P., Croux, C., Robust factor analysis (2003) J Multivariate Anal, 84, pp. 145-172
  • Rodrigues, I.M., (2003) Métodos Robustos em Análise de Componentes Principais Comuns, , http://www.math.ist.utl.pt/~apires/phd.html, Unpublished PhD Thesis (in portuguese), Universidade Técnica de Lisboa
  • Rousseeuw, P.J., Grossmann, W., Pflug, G., Vincze, I., Wertz, W., Multivariate estimation with high breakdown point (1985) Mathematical Statistics and Applications, pp. 283-297. , Reidel Dordrecht
  • Rousseeuw, P.J., Van Zomeren, B.C., Unmasking multivariate outliers and leverage points (1990) J Am Stat Assoc, 85, pp. 633-639
  • Shi, L., Local influence in principal components analysis (1997) Biometrika, 84, pp. 175-186
  • Stahel, W.A., (1981) Robuste Schätzungen: Infinitesimale Optimalität und Schätzungen von Kovarianzmatrizen, , PhD thesis, ETH Zürich
  • Tyler, D., Stahel, W., Weisberg, S., Some issues in the robust estimation of multivariate location and scatter (1991) Directions in Robust Statistics and Diagnostics Part 2, pp. 327-336. , Springer Heidelberg

Citas:

---------- APA ----------
Boente, G., Pires, A.M. & Rodrigues, I.M. (2008) . Estimators for the common principal components model based on reweighting: Influence functions and Monte Carlo study. Metrika, 67(2), 189-218.
http://dx.doi.org/10.1007/s00184-007-0129-4
---------- CHICAGO ----------
Boente, G., Pires, A.M., Rodrigues, I.M. "Estimators for the common principal components model based on reweighting: Influence functions and Monte Carlo study" . Metrika 67, no. 2 (2008) : 189-218.
http://dx.doi.org/10.1007/s00184-007-0129-4
---------- MLA ----------
Boente, G., Pires, A.M., Rodrigues, I.M. "Estimators for the common principal components model based on reweighting: Influence functions and Monte Carlo study" . Metrika, vol. 67, no. 2, 2008, pp. 189-218.
http://dx.doi.org/10.1007/s00184-007-0129-4
---------- VANCOUVER ----------
Boente, G., Pires, A.M., Rodrigues, I.M. Estimators for the common principal components model based on reweighting: Influence functions and Monte Carlo study. Metrika. 2008;67(2):189-218.
http://dx.doi.org/10.1007/s00184-007-0129-4