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The Bennett-Orlicz Norm

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Abstract

van de Geer and Lederer (Probab. Theory Related Fields 157(1-2), 225–250, 2013) introduced a new Orlicz norm, the Bernstein-Orlicz norm, which is connected to Bernstein type inequalities. Here we introduce another Orlicz norm, the Bennett-Orlicz norm, which is connected to Bennett type inequalities. The new Bennett-Orlicz norm yields inequalities for expectations of maxima which are potentially somewhat tighter than those resulting from the Bernstein-Orlicz norm when they are both applicable. We discuss cross connections between these norms, exponential inequalities of the Bernstein, Bennett, and Prokhorov types, and make comparisons with results of Talagrand (Ann. Probab., 17(4), 1546–1570, 1989, 1991), and Boucheron et al. (2013).

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Acknowledgements

I owe thanks to Evan Greene and Johannes Lederer for several helpful conversations and suggestions. Thanks are also due to Richard Nickl for a query concerning Prokhorov’s inequality.

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Authors and Affiliations

  1. Department of Statistics, University of Washington, Seattle, WA, 98195-4322, USA

    Jon A. Wellner

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  1. Jon A. Wellner
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Correspondence to Jon A. Wellner.

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Jon A. Wellner was supported in part by NSF Grants DMS-1104832 and DMS-1566514, and NI-AID grant 2R01 AI291968-04

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Wellner, J.A. The Bennett-Orlicz Norm. Sankhya A 79, 355–383 (2017). https://doi.org/10.1007/s13171-017-0108-4

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