Sankhya A

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The Bennett-Orlicz Norm

Article

Abstract

van de Geer and Lederer (Probab. Theory Related Fields157(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).

Keywords

Bennett’s inequality Exponential bound Maximal inequality Orlicz norm Poisson Prokhorov’s inequality 

AMS (2000) subject classification

Primary: 60E15 60F10 Secondary: 60G50 33E20 

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Copyright information

© Indian Statistical Institute 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WashingtonSeattleUSA

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