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Sharper Sub-Weibull Concentrations

Author

Listed:
  • Huiming Zhang

    (Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China
    Zhuhai UM Science & Technology Research Institute, Zhuhai 519031, China
    These authors contributed equally to this work.)

  • Haoyu Wei

    (Department of Statistics, North Carolina State University, Raleigh, NC 27695, USA
    These authors contributed equally to this work.)

Abstract

Constant-specified and exponential concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics area. We obtain sharper and constants-specified concentration inequalities for the sum of independent sub-Weibull random variables, which leads to a mixture of two tails: sub-Gaussian for small deviations and sub-Weibull for large deviations from the mean. These bounds are new and improve existing bounds with sharper constants. In addition, a new sub-Weibull parameter is also proposed, which enables recovering the tight concentration inequality for a random variable (vector). For statistical applications, we give an ℓ 2 -error of estimated coefficients in negative binomial regressions when the heavy-tailed covariates are sub-Weibull distributed with sparse structures, which is a new result for negative binomial regressions. In applying random matrices, we derive non-asymptotic versions of Bai-Yin’s theorem for sub-Weibull entries with exponential tail bounds. Finally, by demonstrating a sub-Weibull confidence region for a log-truncated Z-estimator without the second-moment condition, we discuss and define the sub-Weibull type robust estimator for independent observations { X i } i = 1 n without exponential-moment conditions.

Suggested Citation

  • Huiming Zhang & Haoyu Wei, 2022. "Sharper Sub-Weibull Concentrations," Mathematics, MDPI, vol. 10(13), pages 1-29, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2252-:d:849019
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    References listed on IDEAS

    as
    1. Shaomin Li & Haoyu Wei & Xiaoyu Lei, 2021. "Heterogeneous Overdispersed Count Data Regressions via Double Penalized Estimations," Papers 2110.03552, arXiv.org, revised Feb 2022.
    2. Shaomin Li & Haoyu Wei & Xiaoyu Lei, 2022. "Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations," Mathematics, MDPI, vol. 10(10), pages 1-25, May.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Haoyu Wei & Runzhe Wan & Lei Shi & Rui Song, 2023. "Zero-Inflated Bandits," Papers 2312.15595, arXiv.org, revised Oct 2024.
    2. Greeshma Balabhadra & El Mehdi Ainasse & Pawel Polak, 2023. "High-Frequency Volatility Estimation with Fast Multiple Change Points Detection," Papers 2303.10550, arXiv.org, revised Jun 2024.
    3. Xiaowei Yang & Xinqiao Liu & Haoyu Wei, 2022. "Concentration inequalities of MLE and robust MLE," Papers 2210.09398, arXiv.org, revised Dec 2022.
    4. Huiming Zhang & Haoyu Wei & Guang Cheng, 2023. "Tight Non-asymptotic Inference via Sub-Gaussian Intrinsic Moment Norm," Papers 2303.07287, arXiv.org, revised Jan 2024.
    5. Tanoue, Yuta, 2024. "Concentration inequality and the weak law of large numbers for the sum of partly negatively dependent φ-subgaussian random variables," Statistics & Probability Letters, Elsevier, vol. 206(C).
    6. Fei Wang & Yuhao Deng, 2023. "Non-Asymptotic Bounds of AIPW Estimators for Means with Missingness at Random," Mathematics, MDPI, vol. 11(4), pages 1-14, February.

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