Concentration inequalities for sampling without replacement

Rémi Bardenet 1 Odalric-Ambrym Maillard 2, 3
2 TAO - Machine Learning and Optimisation
CNRS - Centre National de la Recherche Scientifique : UMR8623, Inria Saclay - Ile de France, UP11 - Université Paris-Sud - Paris 11, LRI - Laboratoire de Recherche en Informatique
Abstract : Concentration inequalities quantify the deviation of a random variable from a fixed value. In spite of numerous applications, such as opinion surveys or ecological counting procedures , few concentration results are known for the setting of sampling without replacement from a finite population. Until now, the best general concentration inequality has been a Hoeffding inequality due to ?. In this paper, we first improve on the fundamental result of ?, and further extend it to obtain a Bernstein concentration bound for sampling without replacement. We then derive an empirical version of our bound that does not require the variance to be known to the user.
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Rémi Bardenet, Odalric-Ambrym Maillard. Concentration inequalities for sampling without replacement. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2015, 21 (3), pp.1361-1385. ⟨10.3150/14-BEJ605⟩. ⟨hal-01216652⟩

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