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

13 Mar 2023  ·  Huiming Zhang, Haoyu Wei, Guang Cheng ·

In non-asymptotic statistical inferences, variance-type parameters of sub-Gaussian distributions play a crucial role. However, direct estimation of these parameters based on the empirical moment generating function (MGF) is infeasible. To this end, we recommend using a sub-Gaussian intrinsic moment norm [Buldygin and Kozachenko (2000), Theorem 1.3] through maximizing a series of normalized moments. Importantly, the recommended norm can not only recover the exponential moment bounds for the corresponding MGFs, but also lead to tighter Hoeffding's sub-Gaussian concentration inequalities. In practice, {\color{black} we propose an intuitive way of checking sub-Gaussian data with a finite sample size by the sub-Gaussian plot}. Intrinsic moment norm can be robustly estimated via a simple plug-in approach. Our theoretical results are applied to non-asymptotic analysis, including the multi-armed bandit.

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