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Asymptotics for Least Absolute Deviation Regression 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 linear regression is defined by minimizing the sum of the absolute values of the residuals. This paper provides a direct proof of asymptotic normality for the LAD estimator. The main theorem assumes deterministic carriers. The extension to random carriers includes the case of autoregressions whose error terms have finite second moments. For a first-order autoregression with Cauchy errors the LAD estimator is shown to converge at a 1/n rate.

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