# The Bennett-Orlicz norm

## TL;DR

This paper introduces the Bennett-Orlicz norm, a new mathematical tool linked to Bennett inequalities, providing potentially tighter bounds for expectations of maxima compared to existing norms.

## Contribution

The paper presents the Bennett-Orlicz norm, expanding the family of Orlicz norms and connecting it to Bennett inequalities for improved expectation bounds.

## Key findings

- Bennett-Orlicz norm yields tighter expectation inequalities.
- Connections established between Bennett-Orlicz, Bernstein, and Prokhorov norms.
- Comparisons made with classical inequalities and prior results.

## Abstract

Lederer and van de Geer (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 (1989, 1994), and Boucheron, Lugosi, and Massart (2013).