Abstract
We consider exact weak and strong Bahadur-Kiefer representations of the least absolute deviation estimator for the linear regression model. The precise behavior of these representations is obtained under minimal conditions.
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References
Arcones, M. A. (1994). Some strong limit theorems for M-estimators, Stochastic Process. Appl., 53, 241–268.
Arcones, M. A. (1995a). Weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions, Stochastic Process. Appl. (submitted).
Arcones, M. A. (1995b). M-estimators converging to a stable limit (preprint).
Arcones, M. A. (1996). The Bahadur-Kiefer representation of L p regression estimators, Econom. Theory, 12, 257–283.
Babu, G. J. (1989). Some strong representations for LAD estimators in linear models, Probab. Theory Related Fields, 83, 547–558.
Bai, Z. D., Chen, X. R., Wu, Y. and Zhao, L. C. (1989). Asymptotic normality of minimum L 1-norm estimates in linear models, Gujarat Statist. Rev., 16, 13–32.
Bassett, G. and Koenker, R. (1978). Asymptotic theory of least absolute error regression, J. Amer. Statist. Assoc., 73, 618–622.
Bloomfield, P. and Steiger, W. L. (1983). Least Absolute Deviations. Theory, Applications, and Algorithms, Birkhäuser, Boston.
Chow, Y. S. and Robbins, H. (1965). On the asymptotic theory of fixed sequential confidence intervals for the mean, Ann. Math. Statist., 36, 457–462.
Davis, R. A., Knight, K. and Liu, J. (1992) M-estimation for antoregression with infinite variance, Stochastic Process. Appl., 40, 145–180.
Draper, N. R. and Smith, H. (1981). Applied Regression Analysis, Wiley, New York.
Giné, E. and Zinn, J. (1986). Lectures on the central limit theorem for empirical processes, Lecture Notes in Math., 1221, 50–112, Springer, New York.
He, X. and Shao, Q-M. (1996). A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs, Ann. Statist., 24, 2608–2630.
Koenker, R. and Portnoy, S. (1987). L-estimation for linear models, J. Amer. Statist. Assoc., 82, 851–857.
Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces, Springer, New York.
Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators, Econom. Theory, 7, 186–199.
Rao, C. R. and Toutenburg, H. (1995). Linear Models, Least Squares and Alternatives, Springer, New York.
Rao, C. R. and Zhao, L. C. (1992). Linear representations of M-estimates in linear models, Canad. J. Statist., 20, 359–368.
Ruppert, D. and Carroll, R. J. (1980). Trimmed least squares estimation in the linear model, J. Amer. Statist. Assoc., 75, 828–838.
Stigler, J. M. (1986). The History of Statistics, Harvard University Press, Cambridge, Massachusetts.
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Arcones, M.A. Second Order Representations of the Least Absolute Deviation Regression Estimator. Annals of the Institute of Statistical Mathematics 50, 87–117 (1998). https://doi.org/10.1023/A:1003401414732
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DOI: https://doi.org/10.1023/A:1003401414732