Sharper Sub-Weibull Concentrations

Huiming Zhang and Haoyu Wei
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Huiming Zhang: Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China
Haoyu Wei: Department of Statistics, North Carolina State University, Raleigh, NC 27695, USA

Mathematics, 2022, vol. 10, issue 13, 1-29

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.

Keywords: constants-specified concentration inequalities; exponential tail bounds; heavy-tailed random variables; sub-Weibull parameter; lower bounds on the least singular value (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (6)

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