Open Access
2012 Concentration inequalities for order statistics
Stéphane Boucheron, Maud Thomas
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Electron. Commun. Probab. 17: 1-12 (2012). DOI: 10.1214/ECP.v17-2210


This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. When the sampling distribution belongs to a maximum domain of attraction, these bounds are checked to be asymptotically tight. When the sampling distribution has a non decreasing hazard rate, we derive an exponential Efron-Stein inequality for order statistics, that is an inequality connecting the logarithmic moment generating function of order statistics with exponential moments of Efron-Stein (jackknife) estimates of variance. This connection is used to derive variance and tail bounds for order statistics of Gaussian samples that are not within the scope of the Gaussian concentration inequality. Proofs are elementary and combine Rényi's representation of order statistics with the entropy approach to concentration of measure popularized by M. Ledoux.


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Stéphane Boucheron. Maud Thomas. "Concentration inequalities for order statistics." Electron. Commun. Probab. 17 1 - 12, 2012.


Accepted: 1 November 2012; Published: 2012
First available in Project Euclid: 7 June 2016

zbMATH: 1349.60021
MathSciNet: MR2994876
Digital Object Identifier: 10.1214/ECP.v17-2210

Primary: 60E15
Secondary: 60F10 , 60G70 , 62G30 , 62G32

Keywords: Concentration inequalities , entropy method , order statistics

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