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Analysis of least absolute deviation

Author

Listed:
  • Kani Chen
  • Zhiliang Ying
  • Hong Zhang
  • Lincheng Zhao

Abstract

We develop a unified L 1 -based analysis-of-variance-type method for testing linear hypotheses. Like the classical L 2 -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. Copyright 2008, Oxford University Press.

Suggested Citation

  • Kani Chen & Zhiliang Ying & Hong Zhang & Lincheng Zhao, 2008. "Analysis of least absolute deviation," Biometrika, Biometrika Trust, vol. 95(1), pages 107-122.
    • Handle: RePEc:oup:biomet:v:95:y:2008:i:1:p:107-122
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    File URL: http://hdl.handle.net/10.1093/biomet/asm082
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    Cited by:

    1. Ke Zhu, 2016. "Bootstrapping the portmanteau tests in weak auto-regressive moving average models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(2), pages 463-485, March.
    2. Guo, Shaojun & Li, Dong & Li, Muyi, 2019. "Strict stationarity testing and GLAD estimation of double autoregressive models," Journal of Econometrics, Elsevier, vol. 211(2), pages 319-337.
    3. Feipeng Zhang & Qunhua Li, 2023. "Segmented correspondence curve regression for quantifying covariate effects on the reproducibility of high‐throughput experiments," Biometrics, The International Biometric Society, vol. 79(3), pages 2272-2285, September.
    4. Ke Zhu & Shiqing Ling, 2015. "LADE-Based Inference for ARMA Models With Unspecified and Heavy-Tailed Heteroscedastic Noises," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 784-794, June.
    5. Nikolaus Hautsch & Julia Schaumburg & Melanie Schienle, 2015. "Financial Network Systemic Risk Contributions," Review of Finance, European Finance Association, vol. 19(2), pages 685-738.
    6. Fumiya Akashi, 2017. "Self-weighted generalized empirical likelihood methods for hypothesis testing in infinite variance ARMA models," Statistical Inference for Stochastic Processes, Springer, vol. 20(3), pages 291-313, October.
    7. Zhu, Ke & Li, Wai Keung, 2015. "A bootstrapped spectral test for adequacy in weak ARMA models," Journal of Econometrics, Elsevier, vol. 187(1), pages 113-130.
    8. Zhouping Li & Yuanyuan Lin & Guoliang Zhou & Wang Zhou, 2014. "Empirical likelihood for least absolute relative error regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 86-99, March.
    9. Ning, Zijun & Tang, Linjun, 2014. "Estimation and test procedures for composite quantile regression with covariates missing at random," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 15-25.
    10. Rong Jiang & Wei-Min Qian & Jing-Ru Li, 2014. "Testing in linear composite quantile regression models," Computational Statistics, Springer, vol. 29(5), pages 1381-1402, October.
    11. Chevapatrakul, Thanaset, 2015. "Monetary environments and stock returns: International evidence based on the quantile regression technique," International Review of Financial Analysis, Elsevier, vol. 38(C), pages 83-108.
    12. Diego R. Moraes Silva & André T. Furtado & Nicholas S. Vonortas, 2018. "University-industry R&D cooperation in Brazil: a sectoral approach," The Journal of Technology Transfer, Springer, vol. 43(2), pages 285-315, April.
    13. Marques, André M. & Lima, Gilberto Tadeu, 2022. "Testing for Granger causality in quantiles between the wage share in income and productive capacity utilization," Structural Change and Economic Dynamics, Elsevier, vol. 62(C), pages 290-312.
    14. Mehmet Hakan Satman, 2022. "Teaching the Median with Terms of Absolute Value, Differentiability, and Optimization," Alphanumeric Journal, Bahadir Fatih Yildirim, vol. 10(1), pages 41-50, June.
    15. Saurabh Bansal & Genaro J. Gutierrez & John R. Keiser, 2017. "Using Experts’ Noisy Quantile Judgments to Quantify Risks: Theory and Application to Agribusiness," Operations Research, INFORMS, vol. 65(5), pages 1115-1130, October.
    16. Noh, Hohsuk & El Ghouch, Anouar & Van Keilegom, Ingrid, 2013. "Assessing model adequacy in possibly misspecified quantile regression," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 558-569.
    17. Chen, Kani & Lin, Yuanyuan & Wang, Zhanfeng & Ying, Zhiliang, 2016. "Least product relative error estimation," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 91-98.
    18. Hui Tian & Andrew Yim & David P. Newton, 2021. "Tail-Heaviness, Asymmetry, and Profitability Forecasting by Quantile Regression," Management Science, INFORMS, vol. 67(8), pages 5209-5233, August.

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