Statistics Surveys

$M$-functionals of multivariate scatter

Lutz Dümbgen, Markus Pauly, and Thomas Schweizer

Full-text: Open access

Abstract

This survey provides a self-contained account of $M$-estimation of multivariate scatter. In particular, we present new proofs for existence of the underlying $M$-functionals and discuss their weak continuity and differentiability. This is done in a rather general framework with matrix-valued random variables. By doing so we reveal a connection between Tyler’s (1987a) $M$-functional of scatter and the estimation of proportional covariance matrices. Moreover, this general framework allows us to treat a new class of scatter estimators, based on symmetrizations of arbitrary order. Finally these results are applied to $M$-estimation of multivariate location and scatter via multivariate $t$-distributions.

Article information

Source
Statist. Surv., Volume 9 (2015), 32-105.

Dates
First available in Project Euclid: 20 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ssu/1426857094

Digital Object Identifier
doi:10.1214/15-SS109

Mathematical Reviews number (MathSciNet)
MR3324616

Zentralblatt MATH identifier
1309.62087

Subjects
Primary: 62G20: Asymptotic properties 62G35: Robustness 62H12: Estimation 62H99: None of the above, but in this section

Keywords
Coercivity convexity matrix exponential function multivariate $t$-distribution scatter functionals weak continuity weak differentiablity

Citation

Dümbgen, Lutz; Pauly, Markus; Schweizer, Thomas. $M$-functionals of multivariate scatter. Statist. Surv. 9 (2015), 32--105. doi:10.1214/15-SS109. https://projecteuclid.org/euclid.ssu/1426857094


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References

  • Arslan, O., Constable, P. D. L. and Kent, J. T. (1995). Convergence behavior of the EM algorithm for the multivariate t-distributions. Communications in Statistics – Theory and Methods 24 2981–3000.
  • Arslan, O. and Kent, J. T. (1998). A note on the maximum likelihoood estimators for the location and scatter parameters of a multivariate Cauchy distribution. Communications in Statistics – Theory and Methods 27 3007–3014.
  • Auderset, C., Mazza, C. and Ruh, E. A. (2005). Angular Gaussian and Cauchy estimation. J. Multivar. Anal. 93 180–197.
  • Bhatia, R. (2007). Positive definite matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ.
  • Croux, C., Rousseeuw, P. J. and Hössjer, O. (1994). Generalized $S$-estimators. J. Amer. Statist. Assoc. 89 1271–1281.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1–38. With discussion.
  • Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist 39 1563–1572.
  • Dudley, R. M. (2002). Real analysis and probability. Cambridge Studies in Advanced Mathematics 74. Cambridge University Press, Cambridge. Revised reprint of the 1989 original.
  • Dudley, R. M., Sidenko, S. and Wang, Z. (2009). Differentiability of $t$-functionals of location and scatter. Ann. Statist. 37 939–960.
  • Dümbgen, L. (1998). On Tyler’s $M$-functional of scatter in high dimension. Ann. Inst. Statist. Math. 50 471–491.
  • Dümbgen, L., Nordhausen, K. and Schuhmacher, H. (2013). New algorithms for $M$-estimation of multivariate scatter and location. ArXiv Preprint, arXiv:1312.6489.
  • Dümbgen, L. and Tyler, D. E. (2005). On the breakdown properties of some multivariate $M$-functionals. Scand. J. Statist. 32 247–264.
  • Eaton, M. L. (1989). Group invariance applications in statistics. NSF-CBMS Regional Conference Series in Probability and Statistics 1. Institute of Mathematical Statistics, Hayward, CA; American Statistical Association, Alexandria, VA.
  • Eriksen, P. S. (1987). Proportionality of covariance matrices. Ann. Statist. 15 732–748.
  • Flury, B. K. (1986). Proportionality of $k$ covariance matrices. Statist. Probab. Lett. 4 29–33.
  • Haberman, S. J. (1989). Concavity and estimation. Ann. Statist. 17 1631–1661.
  • Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York. The approach based on influence functions.
  • Hettmansperger, T. P. and Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika 89 851–860.
  • Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statistics 19 293–325.
  • Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35 73–101.
  • Huber, P. J. (1973). Robust regression: Asymptotics, conjectures and Monte Carlo. Ann. Statist. 1 799–821.
  • Huber, P. J. (1981). Robust statistics. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York.
  • Jensen, S. T. and Johansen, S. (1987). Estimation of proportional covariances. Statist. Probab. Lett. 6 83–85.
  • Kent, J. T. and Tyler, D. E. (1988). Maximum likelihood estimation for the wrapped Cauchy distribution. J. Appl. Statist. 15 247–254.
  • Kent, J. T. and Tyler, D. E. (1991). Redescending $M$-estimates of multivariate location and scatter. Ann. Statist. 19 2102–2119.
  • Kent, J. T., Tyler, D. E. and Vardi, Y. (1994). A curious likelihood identity for the multivariate $t$-distribution. Comm. Statist. Simulation Comput. 23 441–453.
  • Kotz, S. and Nadarajah, S. (2004). Multivariate $t$ distributions and their applications. Cambridge University Press, Cambridge.
  • Lange, K. L., Little, R. J. A. and Taylor, J. M. G. (1989). Robust statistical modeling using the $t$ distribution. J. Amer. Statist. Assoc. 84 881–896.
  • Lehmann, E. L. and Casella, G. (1998). Theory of point estimation, second ed. Springer Texts in Statistics. Springer-Verlag, New York.
  • Maronna, R. A. (1976). Robust $M$-estimators of multivariate location and scatter. Ann. Statist. 4 51–67.
  • Niemiro, W. (1992). Asymptotics for $M$-estimators defined by convex minimization. Ann. Statist. 20 1514–1533.
  • Nordhausen, K., Oja, H. and Ollila, E. (2008). Robust independent component analysis based on two scatter matrices. Australian J. Statist. 37 91–100.
  • Paindaveine, D. (2008). A canonical definition of shape. Statist. Probab. Lett. 78 2240–2247.
  • Serfling, R. J. (1980). Approximation theorems of mathematical statistics. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York.
  • Sirkiä, S., Taskinen, S. and Oja, H. (2007). Symmetrised $M$-estimators of multivariate scatter. J. Multivariate Anal. 98 1611–1629.
  • Skorohod, A. V. (1956). Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen. 1 289–319.
  • Tyler, D. E. (1987a). A distribution-free $M$-estimator of multivariate scatter. Ann. Statist. 15 234–251.
  • Tyler, D. E. (1987b). Statistical analysis for the angular central Gaussian distribution on the sphere. Biometrika 74 579–589.
  • Tyler, D. E., Critchley, F., Dümbgen, L. and Oja, H. (2009). Invariant co-ordinate selection. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 549–592.
  • van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge University Press, Cambridge.
  • Watson, G. S. (1983). Statistics on spheres. University of Arkansas Lecture Notes in the Mathematical Sciences 6. John Wiley & Sons, Inc., New York. A Wiley-Interscience Publication.
  • Wiesel, A. (2012). Geodesic convexity and covariance estimation. IEEE Trans. Signal Process. 60 6182–6189.