M-Estimation and Spatial Quantiles

  • V. Koltchinskii
Part of the Lecture Notes in Statistics book series (LNS, volume 109)


An extension of quantile function (M-quantiles) related in a certain sense to M-parameters of probability distribution defined by convex minimization is considered. Asymtotics of empirical M-quantiles are studied.

Key words and phrases

M-parameter M-estimator quantile function empirical process Bahadur-Kiefer representation 

AMS 1980 subject classifications

60F05 62E20 


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  1. [1]
    Arcones, M.A. and Mason D.M. (1992): A general approach to Bahadur-Kiefer representations for M-estimators. Preprint.Google Scholar
  2. [2]
    Chaudhuri, P. (1993): On a geometric notion of quantiles for multivariate data. Preprint.Google Scholar
  3. [3]
    Dudley, R.M. and Koltchinskii, V. (1992): The spatial quantiles. Preprint.Google Scholar
  4. [4]
    Gine, E. and Zinn, J. (1986): Lectures on the central limit theorem for empirical processes. Lecture Notes in Mathematics #1221 50–112. Springer, New York.Google Scholar
  5. [5]
    Huber, P.J. (1964): Robust estimation of a location parameter. Ann. Math. Statist. 35 73–101.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Huber, P.J. (1967): The behavior of maximum likelihood estimates under non-standard conditions. In Proc. Fifth Berkley Symp. Math. Stat. and Probab. 1 221–223. Univ. of California Press, Berkley.Google Scholar
  7. [7]
    Huber, P.J. (1981): Robust Statistics. John Wiley, New York.CrossRefzbMATHGoogle Scholar
  8. [8]
    Koltchinskii, V. (1994a): Bahadur-Kiefer approximation for spatial quantiles. In Probability in Banach spaces (J. Hoffmann-Jorgensen, J. Kuelbs and M.B. Marcus, eds.), 394–408. Birkhäuser, Boston.Google Scholar
  9. [9]
    Koltchinskii, V. (1994b): Spatial quantiles and their Bahadur-Kiefer representations. In Asymptotic Statistics. P. Mandi and M. Huskova, eds.), 361–367. Physica-Verlag, Vienna.CrossRefGoogle Scholar
  10. [10]
    Koltchinskii, V. (1994c): Nonlinear transformations of empirical processes: functional inverses and Bahadur-Kiefer representations. To appear in Proc. of the 6-th Vilnius Conference on Probability Theory and Mathematical Statistics.Google Scholar
  11. [11]
    Koltchinskii, V. and Dudley, R.M. (1992): Differentiability of inverse operators and asymptotic properties of inverse functions. Preprint.Google Scholar
  12. [12]
    Mattner, L. (1992): Completness of location families, translated moments, and uniqueness of charges. Probab. Th. Rel. Fields 92 137–149.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Pollard, D. (1984): Convergence of Stochastic Processes. Springer-Verlag, New York.CrossRefzbMATHGoogle Scholar
  14. [14]
    Pollard, D. (1990): Empirical Processes: Theory and Applications. Institute of Mathematical Statistics, Hayward, California.zbMATHGoogle Scholar
  15. [15]
    Rockafellar, R.T. (1970): Convex Analysis. Princeton Univ. Press, Princeton.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • V. Koltchinskii
    • 1
  1. 1.University of GiessenGermany

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