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WEIGHTED SAMPLING WITHOUT REPLACEMENT

Abstract : Comparing concentration properties of uniform sampling with and without replacement has a long history which can be traced back to the pioneer work of Hoeffding (1963). The goal of this short note is to extend this comparison to the case of non-uniform weights, using a coupling between the two samples. When the items' weights are arranged in the same order as their values, we show that the induced coupling for the cumulative values is a submartingale coupling. As a consequence, the powerful Chernoff-type upper-tail estimates known for sampling with replacement automatically transfer to the case of sampling without replacement. For general weights, we use the same coupling to establish a sub-Gaussian concentration inequality. We also construct another martingale coupling which allows us to answer a question raised by Luh and Pippenger (2014) on sampling in Polya urns with different replacement numbers.
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https://hal.archives-ouvertes.fr/hal-01376925
Contributor : Anna Ben-Hamou <>
Submitted on : Wednesday, October 5, 2016 - 10:33:43 PM
Last modification on : Friday, March 27, 2020 - 3:01:02 AM
Document(s) archivé(s) le : Friday, January 6, 2017 - 2:44:36 PM

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  • HAL Id : hal-01376925, version 1
  • ARXIV : 1603.06556

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Anna Ben-Hamou, Yuval Peres, Justin Salez. WEIGHTED SAMPLING WITHOUT REPLACEMENT. 2016. ⟨hal-01376925⟩

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