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Asymptotics for Least Absolute DeviationRegression Estimators

Published online by Cambridge University Press:  11 February 2009

David Pollard
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Abstract

The LAD estimator of the vector parameter in a linearregression is defined by minimizing the sum of theabsolute values of the residuals. This paperprovides a direct proof of asymptotic normality forthe LAD estimator. The main theorem assumesdeterministic carriers. The extension to randomcarriers includes the case of autoregressions whoseerror terms have finite second moments. For afirst-order autoregression with Cauchy errors theLAD estimator is shown to converge at a1/n rate.

Information

Type
Research Article
Information
Econometric Theory , Volume 7 , Issue 2 , June 1991 , pp. 186 - 199
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

1. Amemiya, T. Two stage least absolute deviations estimators. Econometrica 50 (1982): 689–711.10.2307/1912608CrossRefGoogle Scholar
2. Amemiya, T. Advanced Econometrics. Cambridge, MA: Harvard University Press, 1985.Google Scholar
3. An, H.-Z. & Chen, Z.-G.. On convergence of LAD estimates in autoregression with infinite variance. Journal of Multivariale Analysis 12 (1982): 335–345.10.1016/0047-259X(82)90070-7CrossRefGoogle Scholar
4. Andersen, P.K. & Gill, R.. Cox's regression model for counting processes: a large sample study. Annals of Statistics 10 (1982): 1100–1120.10.1214/aos/1176345976CrossRefGoogle Scholar
5. Bassett, G. & Koenker, R.. Asymptotic theory of least absolute error regression. Journal of the American Statistical Association 73 (1978): 618–622.10.1080/01621459.1978.10480065Google Scholar
6. Bickel, P.J. One-step Huber estimates in the linear model. J. American Statistical Association 70 (1975): 428–433.10.1080/01621459.1975.10479884Google Scholar
7. Bloomfield, P. & Steiger, W.L.. Least Absolute Deviations: Theory, Applications, and Algorithms. Boston: Birkhauser, 1983.Google Scholar
8. Davis, R.A., Knight, K. & Liu, J.. M-estimation for autoregressions with infinite variance. Preprint (1990).Google Scholar
9. Heiler, S. & Willers, R.. Asymptotic normality of R-estimates in the linear model. Statistics 19 (1988): 173–184.10.1080/02331888808802084CrossRefGoogle Scholar
10. Jureckov´, J. Asymptotic relations of M-estimates and R-estimates in linear regression model. Annals of Statistics 5 (1977): 464–472.Google Scholar
11. Knight, K. A proof of asymptotic normality of LAD and L-estimates in linear regression. Preprint, University of Toronto (1989a).Google Scholar
12. Knight, K. Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics 17 (1989b): 261–278.10.2307/3315522CrossRefGoogle Scholar
13. Pakes, A. & Pollard, D.. Simulation and the asymptotics of optimization estimators. Econometrica 57 (1989): 1027–1058.10.2307/1913622Google Scholar
14. Pollard, D. A central limit theorem for k-means clustering. Annals of Probability 10 (1982): 919–926.10.1214/aop/1176993713Google Scholar
15. Pollard, D. Convergence of Stochastic Processes. New York: Springer-Verlag, 1984.10.1007/978-1-4612-5254-2CrossRefGoogle Scholar
16. Pollard, D. New ways to prove central limit theorems. Econometric Theory. 1 (1985): 295–314.10.1017/S0266466600011233CrossRefGoogle Scholar
17. Pollard, D. Asymptotics via empirical processes. Statistical Science 4 (1989): 341–366.Google Scholar
18. Pollard, D. Empirical processes: theory and applications. NSF-CBMS Regional Conference Series in Probability and Statistics, Vol.2. Hayward, CA: Institute of Mathematical Statistics, 1990.Google Scholar
19. Rockafellar, R.T. Convex Analysis. Princeton, NJ: Princeton University Press, 1970.10.1515/9781400873173Google Scholar
20. Ruppert, D. & Carroll, R.J.. Trimmed least squares estimation in the linear model. J. American Statistical Association 75 (1980): 828–838.10.1080/01621459.1980.10477560Google Scholar
21. Sanz, G. n r-consistency of certain optimal estimators, 0 <r<½ . Preprint, Universidad de Zaragoza, 1988.Google Scholar
22. Van de Geer, S. Asymptotic normality of minimum L 1-norm estimators in linear regression. Preprint, University of Bristol, 1988.Google Scholar