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Concentration inequalities via zero bias couplings

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  • Goldstein, Larry
  • Işlak, Ümit

Abstract

The tails of the distribution of a mean zero, variance σ2 random variable Y satisfy concentration of measure inequalities of the form P(Y≥t)≤exp(−B(t)) forB(t)=t22(σ2+ct)for t≥0,andB(t)=tc(logt−loglogt−σ2c)for t>e whenever there exists a zero biased coupling of Y bounded by c, under suitable conditions on the existence of the moment generating function of Y. These inequalities apply in cases where Y is not a function of independent variables, such as for the Hoeffding statistic Y=∑i=1naiπ(i) where A=(aij)1≤i,j≤n∈Rn×n and the permutation π has the uniform distribution over the symmetric group, and when its distribution is constant on cycle type.

Suggested Citation

  • Goldstein, Larry & Işlak, Ümit, 2014. "Concentration inequalities via zero bias couplings," Statistics & Probability Letters, Elsevier, vol. 86(C), pages 17-23.
    • Handle: RePEc:eee:stapro:v:86:y:2014:i:c:p:17-23
    DOI: 10.1016/j.spl.2013.12.001
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    1. Larry Goldstein & Yosef Rinott, 2003. "A Permutation test for matching and its asymptotic distribution," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 375-388.
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    Cited by:

    1. 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.

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