Skip to main content
SpringerLink
Log in

On Tyler's M-Functional of Scatter in High Dimension

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Let y1, y2,..., yn∈ Rq be independent, identically distributed random vectors with nonsingular covariance matrix Σ, and let S = S(y1,..., yn) be an estimator for Σ. A quantity of particular interest is the condition number of Σ-1 S. If the yi are Gaussian and S is the sample covariance matrix, the condition number of Σ-1 S, i.e. the ratio of its extreme eigenvalues, equals 1 + Op((q/n)1/2) as q →∞ and q/n → 0. The present paper shows that the same result can be achieved with two estimators based on Tyler's (1987, Ann. Statist., 15, 234-251) M-functional of scatter, assuming only elliptical symmetry of ℒ(yi) or less. The main tool is a linear expansion for this M-functional which holds uniformly in the dimension q. As a by-product we obtain continuous Fréchet-differentiability with respect to weak convergence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Arcones, M. A. and Gine, E. (1993). Limit theorems for U-processes, Ann. Probab., 21, 1494–1542.

    Google Scholar 

  • Bai, Z. D. and Wu, Y. (1994a). Limiting behavior of M-estimators of regression coefficients in high dimensional linear models. I. Scale-dependent case, J. Multivariate Anal., 51, 211–239.

    Google Scholar 

  • Bai, Z. D. and Wu, Y. (1994b). Limiting behavior of M-estimators of regression coefficients in high dimensional linear models. II. Scale-invariant case, J. Multivariate Anal., 51, 240–251.

    Google Scholar 

  • Bennett, G. (1962). Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc., 57, 33–45.

    Google Scholar 

  • Bickel, P. and Freedman, D. (1981). Some asymptotic theory for the bootstrap, Ann. Statist., 9, 1196–1217.

    Google Scholar 

  • Clarke, B. R. (1983). Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations, Ann. Statist., 11, 1196–1205.

    Google Scholar 

  • Dümbgen, L. (1997). The asymptotic behavior of Tyler's M-estimator of scatter in high dimension, Beitrag zur Statistik # 23, Heidelberg University.

  • Dümbgen, L. (1998). Perturbation inequalities and confidence sets for functions of a scatter matrix, J. Multivariate Anal., 65, 19–35.

    Google Scholar 

  • Girko, V. L. (1995). Statistical Analysis of Observations of Increasing Dimension, Kluwer, Dordrecht.

    Google Scholar 

  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, 13–30.

    Google Scholar 

  • Huber, P. J. (1981). Robust Statistics, Wiley, New York.

    Google Scholar 

  • Kent, J. T. and Tyler, D. E. (1988). Maximum likelihood estimation for the wrapped Cauchy distribution, Journal of Applied Statistics, 15, 247–254.

    Google Scholar 

  • Kent, J. T. and Tyler, D. E. (1991). Redescending M-estimates of multivariate location and scatter, Ann. Statist., 19, 2102–2119.

    Google Scholar 

  • Mammen, E. (1996). Empirical process of residuals for high-dimensional linear models, Ann. Statist., 24, 307–335.

    Google Scholar 

  • Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter, Ann. Statist., 4, 51–67.

    Google Scholar 

  • Nolan, D. and Pollard, D. (1987). U-processes: rates of convergence, Ann. Statist., 15, 780–799.

    Google Scholar 

  • Pisier, G. (1983). Some applications of the metric entropy condition to harmonic analysis, Banach Spaces, Harmonic Analysis, and Probability Theory (eds. R. C. Blei and S. J. Sidney), Lecture Notes in Math., 995, 123–154, Springer, Berlin.

    Google Scholar 

  • Pollard, D. (1990). Empirical Processes: Theory and Applications, NSF-CBMS Regionial Conference Series in Probability and Statistics, 2, IMS, Hayward, California.

    Google Scholar 

  • Portnoy, S. (1984). Asymptotic behavior M-estimators of p regression parameters when p 2/n is large, I. Consistency, Ann. Statist., 12, 1298–1309.

    Google Scholar 

  • Portnoy, S. (1985). Asymptotic behavior M-estimators of p regression parameters when p 2/n is large, II. Asymptotic normality, Ann. Statist., 13, 1403–1417.

    Google Scholar 

  • Portnoy, S. (1988). Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity, Ann. Statist., 16, 356–366.

    Google Scholar 

  • Silverstein, J. W. (1985). The smallest eigenvalue of a large dimensional Wishart matrix, Ann. Probab., 13, 1364–1368.

    Google Scholar 

  • Tyler, D. E. (1987). A distribution-free M-estimator of multivariate scatter, Ann. Statist., 15, 234–251.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120, Heidelberg, Germany

    Lutz Dümbgen

Authors
  1. Lutz Dümbgen
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Dümbgen, L. On Tyler's M-Functional of Scatter in High Dimension. Annals of the Institute of Statistical Mathematics 50, 471–491 (1998). https://doi.org/10.1023/A:1003573311481

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1003573311481

Search

Navigation