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August 2018 Weighted sampling without replacement
Anna Ben-Hamou, Yuval Peres, Justin Salez
Braz. J. Probab. Stat. 32(3): 657-669 (August 2018). DOI: 10.1214/17-BJPS359

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 note is to extend this comparison to the case of non-uniform weights, using a coupling between samples drawn with and without replacement. 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. As the sample size approaches the total number of items, the variance factor in this inequality displays the same kind of sharpening as Serfling (1974) identified in the case of uniform weights. We also construct an other martingale coupling which allows us to answer a question raised by Luh and Pippenger (2014) on sampling in Polya urns with different replacement numbers.

Citation

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Anna Ben-Hamou. Yuval Peres. Justin Salez. "Weighted sampling without replacement." Braz. J. Probab. Stat. 32 (3) 657 - 669, August 2018. https://doi.org/10.1214/17-BJPS359

Information

Received: 1 April 2016; Accepted: 1 March 2017; Published: August 2018
First available in Project Euclid: 8 June 2018

zbMATH: 06930043
MathSciNet: MR3812386
Digital Object Identifier: 10.1214/17-BJPS359

Rights: Copyright © 2018 Brazilian Statistical Association

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Vol.32 • No. 3 • August 2018
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