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Tight Non-asymptotic Inference via Sub-Gaussian Intrinsic Moment Norm

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  • Huiming Zhang
  • Haoyu Wei
  • Guang Cheng

Abstract

In non-asymptotic learning, variance-type parameters of sub-Gaussian distributions are of paramount importance. However, directly estimating these parameters using the empirical moment generating function (MGF) is infeasible. To address this, we suggest using the sub-Gaussian intrinsic moment norm [Buldygin and Kozachenko (2000), Theorem 1.3] achieved by maximizing a sequence of normalized moments. Significantly, the suggested norm can not only reconstruct the exponential moment bounds of MGFs but also provide tighter sub-Gaussian concentration inequalities. In practice, we provide an intuitive method for assessing whether data with a finite sample size is sub-Gaussian, utilizing the sub-Gaussian plot. The intrinsic moment norm can be robustly estimated via a simple plug-in approach. Our theoretical findings are also applicable to reinforcement learning, including the multi-armed bandit scenario.

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  • 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.
    • Handle: RePEc:arx:papers:2303.07287
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    References listed on IDEAS

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    1. Timothy B. Armstrong & Michal Kolesár, 2021. "Finite‐Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness," Econometrica, Econometric Society, vol. 89(3), pages 1141-1177, May.
    2. Sylvain Chassang, 2009. "Non-asymptotic tests of model performance," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 41(3), pages 495-514, December.
    3. Huiming Zhang & Haoyu Wei, 2022. "Sharper Sub-Weibull Concentrations," Mathematics, MDPI, vol. 10(13), pages 1-29, June.
    4. Rousseeuw, Peter J. & Verboven, Sabine, 2002. "Robust estimation in very small samples," Computational Statistics & Data Analysis, Elsevier, vol. 40(4), pages 741-758, October.
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