On Tyler's M-functional of scatter in high dimension

Annals of the Institute of Statistical Mathematics

Volumn 50, Issue 3, 1998, Pages 471-491

  Dümbgen, Lutz ^a,b  

^a UNIVERSITY OF HEIDELBERG   (Germany)

^b UNIVERSITY OF LÜBECK   (Germany)

Author keywords

Differentiability; Dimensional asymptotics; Elliptical symmetry; M functional; Scatter matrix; Symmetrization

Indexed keywords

EID: 7744247580     PISSN: 00203157     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1003573311481     Document Type: Article

References (23)

1. Arcones, M. A. and Gine, E. (1993). Limit theorems for U-processes, Ann. Probab., 21, 1494-1542.
2. 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.
3. 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.
4. Bennett, G. (1962). Probability inequalities for the sum of independent random variables, J. Amer. Statist. Assoc., 57, 33-45.
5. Bickel, P. and Freedman, D. (1981). Some asymptotic theory for the bootstrap, Ann. Statist., 9, 1196-1217.
6. Clarke, B. R. (1983). Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations, Ann. Statist., 11, 1196-1205.
7. Dümbgen, L. (1997). The asymptotic behavior of Tyler's M-estimator of scatter in high dimension, Beitrag zur Statistik # 23, Heidelberg University.
8. Dümbgen, L. (1998). Perturbation inequalities and confidence sets for functions of a scatter matrix, J. Multivariate Anal., 65, 19-35.
9. Girko, V. L. (1995). Statistical Analysis of Observations of Increasing Dimension, Kluwer, Dordrecht.
10. Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, 13-30.
11. Huber, P. J. (1981). Robust Statistics, Wiley, New York.
12. Kent, J. T. and Tyler, D. E. (1988). Maximum likelihood estimation for the wrapped Cauchy distribution, Journal of Applied Statistics, 15, 247-254.
13. Kent, J. T. and Tyler, D. E. (1991). Redescending M-estimates of multivariate location and scatter, Ann. Statist., 19, 2102-2119.
14. Mammen, E. (1996). Empirical process of residuals for high-dimensional linear models, Ann. Statist., 24, 307-335.
15. Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter, Ann. Statist., 4, 51-67.
16. Nolan, D. and Pollard, D. (1987). U-processes: rates of convergence, Ann. Statist., 15, 780-799.
17. 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.
18. Pollard, D. (1990). Empirical Processes: Theory and Applications, NSF-CBMS Regionial Conference Series in Probability and Statistics, 2, IMS, Hayward, California.
19. 2/n is large, I. Consistency, Ann. Statist., 12, 1298-1309.
20. 2/n is large, II. Asymptotic normality, Ann. Statist., 13, 1403-1417.
21. Portnoy, S. (1988). Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity, Ann. Statist., 16, 356-366.
22. Silverstein, J. W. (1985). The smallest eigenvalue of a large dimensional Wishart matrix, Ann. Probab., 13, 1364-1368.
23. Tyler, D. E. (1987). A distribution-free M-estimator of multivariate scatter, Ann. Statist., 15, 234-251.