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Asymptotics for argmin processes: Convexity arguments

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  • Kato, Kengo

Abstract

The convexity arguments developed by Pollard [D. Pollard, Asymptotics for least absolute deviation regression estimators, Econometric Theory 7 (1991) 186-199], Hjort and Pollard [N.L. Hjort, D. Pollard, Asymptotics for minimizers of convex processes, 1993 (unpublished manuscript)], and Geyer [C.J. Geyer, On the asymptotics of convex stochastic optimization, 1996 (unpublished manuscript)] are now basic tools for investigating the asymptotic behavior of M-estimators with non-differentiable convex objective functions. This paper extends the scope of convexity arguments to the case where estimators are obtained as stochastic processes. Our convexity arguments provide a simple proof for the asymptotic distribution of regression quantile processes. In addition to quantile regression, we apply our technique to LAD (least absolute deviation) inference for threshold regression.

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  • Kato, Kengo, 2009. "Asymptotics for argmin processes: Convexity arguments," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1816-1829, September.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:8:p:1816-1829
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    6. Stefano Maria IACUS, 2010. "On Lasso-type estimation for dynamical systems with small noise," Departmental Working Papers 2010-12, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.
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    9. Alexis Derumigny & Jean-David Fermanian, 2018. "About Kendall's regression," Working Papers 2018-01, Center for Research in Economics and Statistics.
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    11. Han, Heejoon & Linton, Oliver & Oka, Tatsushi & Whang, Yoon-Jae, 2016. "The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series," Journal of Econometrics, Elsevier, vol. 193(1), pages 251-270.
    12. Chung-Ming Kuan & Christos Michalopoulos & Zhijie Xiao, 2017. "Quantile Regression on Quantile Ranges – A Threshold Approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(1), pages 99-119, January.
    13. Khai X. Chiong & Hyungsik Roger Moon, 2017. "Estimation of Graphical Models using the $L_{1,2}$ Norm," Papers 1709.10038, arXiv.org, revised Oct 2017.
    14. Murat Genç, 2022. "A new double-regularized regression using Liu and lasso regularization," Computational Statistics, Springer, vol. 37(1), pages 159-227, March.
    15. Derumigny, Alexis & Fermanian, Jean-David, 2020. "On Kendall’s regression," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    16. Derumigny, Alexis & Fermanian, Jean-David, 2019. "A classification point-of-view about conditional Kendall’s tau," Computational Statistics & Data Analysis, Elsevier, vol. 135(C), pages 70-94.
    17. Wenjie Wang & Yichong Zhang, 2021. "Wild Bootstrap for Instrumental Variables Regressions with Weak and Few Clusters," Papers 2108.13707, arXiv.org, revised Jan 2024.
    18. Christis Katsouris, 2023. "Quantile Time Series Regression Models Revisited," Papers 2308.06617, arXiv.org, revised Aug 2023.
    19. Benjamin Poignard, 2020. "Asymptotic theory of the adaptive Sparse Group Lasso," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(1), pages 297-328, February.
    20. Christis Katsouris, 2022. "Asymptotic Theory for Unit Root Moderate Deviations in Quantile Autoregressions and Predictive Regressions," Papers 2204.02073, arXiv.org, revised Aug 2023.
    21. Bucher, Axel & El Ghouch, Anouar & Van Keilegom, Ingrid, 2014. "Single-index quantile regression models for censored data," LIDAM Discussion Papers ISBA 2014001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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