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Limiting distributions for \(L_1\) regression estimators under general conditions. (English) Zbl 0929.62021
Summary: It is well known that \(L_1\)-estimators of regression parameters are asymptotically normal if the distribution function has a positive derivative at 0. We derive the asymptotic distributions under more general conditions on the behavior of the distribution function near 0.

MSC:
62F12 Asymptotic properties of parametric estimators
62J05 Linear regression; mixed models
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
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[1] Akahira, M. and Takeuchi, K. (1981). Asy mptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asy mptotic Efficiency. Springer, New York. · Zbl 0463.62026
[2] Akahira, M. and Takeuchi, K. (1995). Non-Regular Statistical Estimation. Springer, New York. · Zbl 0833.62024
[3] Arcones, M. A. (1994). Distributional convergence of M-estimators under unusual rates. Statist. Probab. Lett. 21 271-280. Arcones, M. A. (1996a). The Bahadur-Kiefer representation of Lp regression estimators. Econometric Theory 12 257-283. Arcones, M. A. (1996b). Second order representations of the least absolute deviation regression estimator. Unpublished manuscript. · Zbl 0815.62011 · doi:10.1016/0167-7152(94)00013-1
[4] Babu, G. J. (1989). Strong representation for LAD estimators in linear models. Probab. Theory Related Fields 83 547-558. · Zbl 0665.62033 · doi:10.1007/BF01845702
[5] Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577-580. · Zbl 0147.18805 · doi:10.1214/aoms/1177699450
[6] Bai, Z. D., Chen, X. R., Wu, Y. and Zhao, L. C. (1990). Asy mptotic normality of minimum L1 norm estimates in linear models. Chinese Sciences A 33 449-463.
[7] Bassett, G. and Koenker, R. (1978). Asy mptotic theory of least absolute error regression. J. Amer. Statist. Assoc. 73 618-622. · Zbl 0391.62046 · doi:10.2307/2286611
[8] Bloomfield, P. and Steiger, W. L. (1983). Least Absolute Deviations: Theory, Applications and Algorithms. Birkhäuser, Boston. · Zbl 0536.62049
[9] Gey er, C. J. (1996). On the asy mptotics of convex stochastic optimization. Unpublished manuscript.
[10] He, X. and Shao, Q.-M. (1996). General Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs. Ann. Statist. 24 2608-2630. · Zbl 0867.62012 · doi:10.1214/aos/1032181172
[11] Hjørt, N. L. and Pollard, D. (1993). Asy mptotics for minimisers of convex processes. Statistical Research Report, Univ. Oslo.
[12] Ibragimov, I. A. and Has’minskii, R. Z. (1981). Statistical Estimation: Asy mptotic Theory. Springer, New York.
[13] Jure cková, J. (1983). Asy mptotic behavior of M-estimators of location in nonregular cases. Statist. Decisions 1 323-340. · Zbl 0548.62025
[14] Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie, Berlin. · Zbl 0544.60053
[15] Kiefer, J. (1967). On Bahadur’s representation of sample quantiles. Ann. Math. Statist. 38 1323- 1342. · Zbl 0158.37005 · doi:10.1214/aoms/1177698690
[16] Knight, K. (1997). Asy mptotics for L1 regression estimators under general conditions. Technical Report 9716, Dept. Statistics, Univ. Toronto. · Zbl 0935.62095 · doi:10.1214/lnms/1215454147
[17] Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33-50. JSTOR: · Zbl 0373.62038 · doi:10.2307/1913643 · links.jstor.org
[18] Pollard, D. (1991). Asy mptotics for least absolute deviation regression estimators. Econometric Theory 7 186-199. JSTOR: · Zbl 04504753 · doi:10.1017/S0266466600004394 · links.jstor.org
[19] Smirnov, N. V. (1952). Limit distributions for the terms of a variational series. Amer. Math. Soc. Transl. Ser. (1) 11 82-143.
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