Skip Navigation


Biometrika Advance Access originally published online on February 4, 2008
Biometrika 2008 95(1):107-122; doi:10.1093/biomet/asm082
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Chen, K.
Right arrow Articles by Zhao, L.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2008 Biometrika Trust

Articles

Analysis of least absolute deviation

Kani Chen

Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong makchen{at}ust.hk

Zhiliang Ying

Department of Statistics, Columbia University, New York, New York 10027, U.S.A. zying{at}stat.columbia.edu

Hong Zhang and Lincheng Zhao

Department of Statistics and Finance, University of Science and Technology of China, Heifei, Anhui 230026, China zhangh{at}ustc.edu.cn lczhao{at}ustc.edu.cn

Received for publication 1 March 2006. Revision received 1 June 2007.

We develop a unified L1-based analysis-of-variance-type method for testing linear hypotheses. Like the classical L2-based analysis of variance, the method is coordinate-free in the sense that it is invariant under any linear transformation of the covariates or regression parameters. Moreover, it allows singular design matrices and heterogeneous error terms. A simple approximation using stochastic perturbation is proposed to obtain cut-off values for the resulting test statistics. Both test statistics and distributional approximations can be computed using standard linear programming. An asymptotic theory is derived for the method. Special cases of one- and multi-way analysis of variance and analysis of covariance models are worked out in detail. The main results of this paper can be extended to general quantile regression. Extensive simulations show that the method works well in practical settings. The method is also applied to a dataset from General Social Surveys.

Key Words: Analysis of covariance • Analysis of variance • Asymptotic expansion • Distributional approximation • Factorial design • Linear constraint • Linear hypothesis • Linear programming • Linear regression • One-way layout • Random perturbation • Quantile regression



References

    Bassett G., Koenker R. Asymptotic theory of least absolute error regression. J. Am. Statist. Assoc. (1978) 73:618–22.[CrossRef][Web of Science]

    Buchinsky M. Recent advances in quantile regression models: a practical guideline for empirical research. J. Hum. Res. (1998) 33:88–126.[CrossRef][Web of Science]

    Charnes A., Cooper W. W., Ferguson R. O. Optimal estimation of executive compensation by linear programming. Manag. Sci. (1955) 1:138–51.

    Chatterjee S., Bose A. Generalized bootstrap for estimating equations. Ann. Statist. (2005) 33:414–36.[CrossRef]

    Chen X. R., Bai Z. D., Zhao L. C., Wu Y. H. Asymptotic normality of minimum L1-norm estimates in linear models. Sci. China (1990) 33:1311–28.

    Freedman D. A., Peters S. C. Bootstrapping a regression equation: some empirical results. J. Am. Statist. Assoc. (1984) 79:97–106.[CrossRef][Web of Science]

    Gutenbrunner C., Jureckova J. Regression rank scores and regression quantiles. Ann. Statist. (1992) 20:305–30.[CrossRef]

    Gutenbrunner C., Jureckova J., Koenker R., Portnoy S. Tests of linear hypotheses based on regression rank scores. J. Nonparam. Statist. (1993) 2:307–31.[CrossRef]

    Jin Z., Ying Z., Wei L. J. A simple resampling method by perturbing the minimand. Biometrika (2001) 88:381–90.[Abstract/Free Full Text]

    Koenker R. Quantile Regression. (2005) Cambridge: Cambridge University Press.

    Koenker R., Bassett G. Regression quantiles. Econometrica (1978) 46:33–50.[CrossRef][Web of Science]

    Koenker R., Bassett G. Robust tests for heteroscedasticity based on regression quantiles. Econometrica (1982) 50:43–61.[CrossRef][Web of Science]

    Koenker R., D'Orey V. Computing regression quantiles. Appl. Statist. (1987) 36:383–93.[CrossRef]

    Parzen M. I., Wei L. J., Ying Z. A resampling method based on pivotal estimating functions. Biometrika (1994) 81:341–50.[Abstract/Free Full Text]

    Portnoy S., Koenker R. The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators (with Discussion). Statist. Sci. (1997) 12:279–300.[CrossRef]

    Pollard D. Asymptotics for least absolute deviations regression estimators. Economet. Theory (1991) 7:186–99.

    Powell J. L. Least absolute deviations estimation for the censored regression model. J. Economet. (1984) 25:303–25.[CrossRef]

    Rao C. R., Zhao L. C. Approximation to the distribution of M-estimates in linear models by randomly weighted bootstrap. Sankhya A (1992) 54:323–31.

    Scheffé H. The Analysis of Variance. (1959) New York: Wiley.

    Ying Z., Jung S. H., Wei L. J. Survival analysis with median regression model. J. Am. Statist. Assoc. (1995) 90:178–84.[CrossRef][Web of Science]


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Chen, K.
Right arrow Articles by Zhao, L.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?