On the uniqueness of S-functionals and M-functionals under nonelliptical distributions

Kay S. Tatsuoka, David E. Tyler

Research output: Contribution to journalArticlepeer-review

86 Scopus citations


The S-functionals of multivariate location and scatter, including the MVE-functionals, are known to be uniquely defined only at unimodal elliptically symmetric distributions. The goal of this paper is to establish the uniqueness of these functionals under broader classes of symmetric distributions. We also discuss some implications of the uniqueness of the functionals and give examples of striclty unimodal and symmetric distributions for which the MVE-functional is not uniquely defined. The uniqueness results for the S-functionals are obtained by embedding them within a more general class of functionals which we call the M-functionals with auxiliary scale. The uniqueness results of this paper are then obtained for this class of multivariate functionals. Besides the S-functionals, the class of multivariate M-functionals with auxiliary scale include the constrained M-functionals recently introduced by Kent and Tyler, as well as a new multivariate generalization of Yohai's MM-functionals.

Original languageEnglish (US)
Pages (from-to)1219-1243
Number of pages25
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 2000

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • CM-estimates
  • Elliptical distributions
  • M-estimates
  • MM-estimates
  • Majorization
  • Minimum volume ellipsoid
  • Permutation invariance
  • Robustness
  • S-estimates
  • Schur-concavity
  • Symmetric exchangeable distributions
  • Unimodal distributions


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