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Biometrika Advance Access originally published online on February 4, 2008
Biometrika 2008 95(1):107-122; doi:10.1093/biomet/asm082
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© 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.
  Abstract

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


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