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The Bennett-Orlicz Norm

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  • Jon A. Wellner

    (University of Washington)

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).

Suggested Citation

  • Jon A. Wellner, 2017. "The Bennett-Orlicz Norm," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 355-383, August.
    • Handle: RePEc:spr:sankha:v:79:y:2017:i:2:d:10.1007_s13171-017-0108-4
    DOI: 10.1007/s13171-017-0108-4
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    References listed on IDEAS

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    1. Goldstein, Larry & Işlak, Ümit, 2014. "Concentration inequalities via zero bias couplings," Statistics & Probability Letters, Elsevier, vol. 86(C), pages 17-23.
    2. Arcones, Miguel A. & Giné, Evarist, 1995. "On the law of the iterated logarithm for canonical U-statistics and processes," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 217-245, August.
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    Cited by:

    1. Jacob Bien & Irina Gaynanova & Johannes Lederer & Christian L. Müller, 2019. "Prediction error bounds for linear regression with the TREX," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 451-474, June.

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