Concentration inequalities for order statistics

Abstract : This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those bounds are checked to be asymptotically tight when the sampling distribution belongs to a maximum domain of attraction. If the sampling distribution has non-decreasing hazard rate (this includes the Gaussian distribution), we derive an exponential Efron-Stein inequality for order statistics: an inequality connecting the logarithmic moment generating function of centered order statistics with exponential moments of Efron-Stein (jackknife) estimates of variance. We use this general connection to derive variance and tail bounds for order statistics of Gaussian sample. Those bounds are not within the scope of the Tsirelson-Ibragimov-Sudakov Gaussian concentration inequality. Proofs are elementary and combine Rényi's representation of order statistics and the so-called entropy approach to concentration inequalities popularized by M. Ledoux.
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Article dans une revue
Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2012, 17, pp.51. <10.1214/ECP.v17-2210>
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https://hal.archives-ouvertes.fr/hal-00751496
Contributeur : Stephane Boucheron <>
Soumis le : mardi 13 novembre 2012 - 15:38:45
Dernière modification le : mardi 11 octobre 2016 - 14:09:12

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UPMC | INSMI | PMA | USPC

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Stephane Boucheron, Maud Thomas. Concentration inequalities for order statistics. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2012, 17, pp.51. <10.1214/ECP.v17-2210>. <hal-00751496>

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