The Annals of Statistics

Bahadur representation of $M\sb m$ estimates

Arup Bose

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We take a unified approach to asymptotic properties of $M_m$ estimates based on i.i.d. observations defined through the minimization of a real-valued criterion function of one or more variables. Our results are applicable to a host of location and scale estimators found in the literature.

Article information

Ann. Statist., Volume 26, Number 2 (1998), 771-777.

First available in Project Euclid: 31 July 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations 60F15: Strong theorems 62E20: Asymptotic distribution theory 62F10: Point estimation 62G30: Order statistics; empirical distribution functions 62G35: Robustness 62H10: Distribution of statistics 62H12: Estimation 62J05: Linear regression

$M$ estimates $U$ statistics asymptotic normality Bahadur representation measures of location measures of dispersion $L^1$ median Oja median $L^t$ estimates Hodges-Lehmann estimate generalized order statistics


Bose, Arup. Bahadur representation of $M\sb m$ estimates. Ann. Statist. 26 (1998), no. 2, 771--777. doi:10.1214/aos/1028144859.

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  • Arcones, M. A. (1996). The Bahadur-Kiefer representation for U-quantiles. Ann. Statist. 24 1400-1422.
  • Arcones, M. A., Chen, Z. and Gin´e, E. (1994). Estimators related to U-processes with applications to multivariate medians: asy mptotic normality. Ann. Statist. 22 1460-1477.
  • Arcones, M. A. and Mason, D. M. (1992). A general approach to the Bahadur-Keifer representations for M-estimators. Technical Report 054-92, Mathematical Sciences Research Institute, Berkeley.
  • Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577-580.
  • Bickel, P. J. and Lehmann, E. L. (1979). Descriptive statistics for nonparametric models. IV. Spread. In Contributions to Statistics (J. Jure ckov´a, ed.) 33-40. Academia, Prague.
  • Bose, A. (1997). Bahadur representation and other asy mptotic properties of M estimates based on U functionals. Technical report, Stat-Math Unit, ISI, Calcutta.
  • Chaudhuri, P. (1992). Multivariate location estimation using extension of R-estimates through U-statistics ty pe approach. Ann. Statist. 20 897-916.
  • Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91 862-872.
  • Choudhury, J. and Serfling, R. J. (1988). Generalized order statistics, Bahadur representations, and sequential nonparametric fixed-width confidence intervals. J. Statist. Plann. Inference 19 269-282.
  • Davies, L. (1992). The asy mptotics of Rouseeuw's minimum volume ellipsoid estimator. Ann. Statist. 20 1828-1843.
  • Habermann, S. J. (1989). Concavity and estimation. Ann. Statist. 17 1631-1661.
  • He, X. and Shao, Q. M. (1996). A general Bahadur representaion of M-estimators and its application to linear regression with nonstochastic designs. Ann. Statist. 24 2608-2630.
  • He, X. and Wang, G. (1995). Law of iterated logarithm and invariance principle for M-estimators. Proc. Amer. Math. Soc. 123 563-573.
  • Heiler, S. and Willers, R. (1988). Asy mptotic normality of R-estimates in the linear model. Statistics 19 173-184.
  • Hjort, N. L. and Pollard, D. (1993). Asy mptotics for minimizers of convex processes. Technical report, Univ. Oslo.
  • Hollander, M. and Wolfe, D. A. (1973). Nonparametrical Statistical Methods. Wiley, New York.
  • Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35 73-101.
  • Jure ckova, J. (1977). Asy mptotic relation of M-estimates and R-estimates in linear regression models. Ann. Statist. 5 464-472.
  • Kiefer, J. (1967). On Bahadur's representation of sample quantiles. Ann. Math. Statist. 38 1323- 1342.
  • Kim, J. and Pollard, D. (1990). Cube root asy mptotics. Ann. Statist. 18 191-219.
  • Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405-414.
  • Maritz, J. S., Wu, M. and Staudte, R. G. (1977). A location estimator based on U statistic. Ann. Statist. 5 779-786.
  • Niemiro, W. (1992). Asy mptotics for M-estimators defined by convex minimization. Ann. Statist. 20 1514-1533.
  • Oja, H. (1983). Descriptive statistics for multivariate distribution. Statist. Probab. Lett. 1 327- 333.
  • Oja, H. (1984). Asy mptotical properties of estimators based on U statistics. Preprint, Dept. Applied Mathematics, Univ. Oulu, Finland.
  • Pollard, D. (1991). Asy mptotics for least absolute deviation regression estimators. Econometric Theory 7 186-199.
  • Rousseeuw, P. J. (1986). Multivariate estimation with high breakdown point. In Mathematical Statistics and Applications (W. Grossman, G. Pflng, I. Vinage and W. Wertz, eds.) 283- 297. Reidel, Dordrecht.
  • Tukey, J. W. (1975). Mathematics and the filtering of data. In Proceedings of the International Congress of Mathematicians, Vancouver, 1974 2 523-531.