Sub-Gaussian Error Bounds for Hypothesis Testing

2021/01/01 by Yan Wang, Wang, Yan
Computer Science · Mathematics · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Information Theory (cs.IT) #Statistical Mechanics and Entropy #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2101.00136

openalex publication_date 2021/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We interpret likelihood-based test functions from a geometric perspective where the Kullback-Leibler (KL) divergence is adopted to quantify the distance from a distribution to another. Such a test function can be seen as a sub-Gaussian random variable, and we propose a principled way to calculate its corresponding sub-Gaussian norm. Then an error bound for binary hypothesis testing can be obtained in terms of the sub-Gaussian norm and the KL divergence, which is more informative than Pinsker's bound when the significance level is prescribed. For M-ary hypothesis testing, we also derive an error bound which is complementary to Fano's inequality by being more informative when the number of hypotheses or the sample size is not large.

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