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Concentration inequalities for order statistics. (English) Zbl 1349.60021
Summary: 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 [The concentration of measure phenomenon. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0995.60002)].

MSC:
60E15 Inequalities; stochastic orderings
60G70 Extreme value theory; extremal stochastic processes
62G30 Order statistics; empirical distribution functions
62G32 Statistics of extreme values; tail inference
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