On the breakdown properties of some multivariate M-functionals

Lutz Dümbgen, David E. Tyler

Research output: Contribution to journalReview articlepeer-review

35 Scopus citations


For probability distributions on R q, a detailed study of the breakdown properties of some multivariate M-functionals related to Tyler's [Ann. Statist. 15 (1987) 234] 'distribution-free' M-functional of scatter is given. These include a symmetrized version of Tyler's M-functional of scatter, and the multivariate t M-functionals of location and scatter. It is shown that for 'smooth' distributions, the (contamination) breakdown point of Tyler's M-functional of scatter and of its symmetrized version are 1/q and 1 - √/1-1/q, respectively. For the multivariate t M-functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on v ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/(q + v). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low-dimensional subspaces.

Original languageEnglish (US)
Pages (from-to)247-264
Number of pages18
JournalScandinavian Journal of Statistics
Issue number2
StatePublished - Jun 2005

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Breakdown
  • Coplanar contamination
  • M-estimates
  • M-functionals
  • Scatter matrix
  • Symmetrization
  • T-distributions
  • Tight contamination


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