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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7 (1991) 186-199], Hjort and Pollard [N.L. Hjort, D. Pollard, Asymptotics for minimizers of convex processes, 1993 (unpublished manuscript)], and Geyer [C.J. Geyer, On the asymptotics of convex stochastic optimization, 1996 (unpublished manuscript)] are now basic tools for investigating the asymptotic behavior of M-estimators with non-differentiable convex objective functions.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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    Cited by:

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    5. Zhang, Feipeng & Xu, Yixiong & Fan, Caiyun, 2023. "Nonparametric inference of expectile-based value-at-risk for financial time series with application to risk assessment," International Review of Financial Analysis, Elsevier, vol. 90(C).
    6. Christis Katsouris, 2023. "Quantile Time Series Regression Models Revisited," Papers 2308.06617, arXiv.org, revised Aug 2023.
    7. 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.
    8. 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).
    9. Jiang, Liang & Phillips, Peter C.B. & Tao, Yubo & Zhang, Yichong, 2023. "Regression-adjusted estimation of quantile treatment effects under covariate-adaptive randomizations," Journal of Econometrics, Elsevier, vol. 234(2), pages 758-776.
    10. Parker, Thomas, 2019. "Asymptotic inference for the constrained quantile regression process," Journal of Econometrics, Elsevier, vol. 213(1), pages 174-189.
    11. Wang, Yunyun & Oka, Tatsushi & Zhu, Dan, 2023. "Bivariate distribution regression with application to insurance data," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 215-232.
    12. Liang Jiang & Xiaobin Liu & Peter C. B. Phillips & Yichong Zhang, 2024. "Bootstrap Inference for Quantile Treatment Effects in Randomized Experiments with Matched Pairs," The Review of Economics and Statistics, MIT Press, vol. 106(2), pages 542-556, March.
    13. Tatsushi Oka & Shota Yasui & Yuta Hayakawa & Undral Byambadalai, 2024. "Regression Adjustment for Estimating Distributional Treatment Effects in Randomized Controlled Trials," Papers 2407.14074, arXiv.org.
    14. Ke Zhu, 2018. "Statistical inference for autoregressive models under heteroscedasticity of unknown form," Papers 1804.02348, arXiv.org, revised Aug 2018.
    15. 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.
    16. Wenjie Wang & Yichong Zhang, 2024. "Gradient Wild Bootstrap for Instrumental Variable Quantile Regressions with Weak and Few Clusters," Papers 2408.10686, arXiv.org.
    17. Yunyun Wang & Tatsushi Oka & Dan Zhu, 2023. "Distributional Vector Autoregression: Eliciting Macro and Financial Dependence," Papers 2303.04994, arXiv.org.
    18. Wenjie Wang & Yichong Zhang, 2021. "Wild Bootstrap for Instrumental Variables Regressions with Weak and Few Clusters," Papers 2108.13707, arXiv.org, revised Jan 2024.
    19. Christis Katsouris, 2022. "Asymptotic Theory for Unit Root Moderate Deviations in Quantile Autoregressions and Predictive Regressions," Papers 2204.02073, arXiv.org, revised Aug 2023.
    20. Bianchi, Pascal & Elgui, Kevin & Portier, François, 2023. "Conditional independence testing via weighted partial copulas," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    21. Christis Katsouris, 2023. "Estimating Conditional Value-at-Risk with Nonstationary Quantile Predictive Regression Models," Papers 2311.08218, arXiv.org, revised Apr 2024.
    22. Alexis Derumigny & Jean-David Fermanian, 2018. "About Kendall's regression," Working Papers 2018-01, Center for Research in Economics and Statistics.
    23. Murat Genç, 2022. "A new double-regularized regression using Liu and lasso regularization," Computational Statistics, Springer, vol. 37(1), pages 159-227, March.
    24. 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.
    25. 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.

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