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November 2019 Bernstein-type exponential inequalities in survey sampling: Conditional Poisson sampling schemes
Patrice Bertail, Stephan Clémençon
Bernoulli 25(4B): 3527-3554 (November 2019). DOI: 10.3150/18-BEJ1101


This paper is devoted to establishing exponential bounds for the probabilities of deviation of a sample sum from its expectation, when the variables involved in the summation are obtained by sampling in a finite population according to a rejective scheme, generalizing simple random sampling without replacement, and by using an appropriate normalization. In contrast to Poisson sampling, classical deviation inequalities in the i.i.d. setting do not straightforwardly apply to sample sums related to rejective schemes, due to the inherent dependence structure of the sampled points. We show here how to overcome this difficulty, by combining the formulation of rejective sampling as Poisson sampling conditioned upon the sample size with the Esscher transformation. In particular, the Bennett/Bernstein type bounds thus established highlight the effect of the asymptotic variance of the (properly standardized) sample weighted sum and are shown to be much more accurate than those based on the negative association property shared by the terms involved in the summation. Beyond its interest in itself, such a result for rejective sampling is crucial, insofar as it permit to obtain tail bounds for many other sampling schemes, namely those that can be accurately approximated by rejective plans in the sense of the total variation distance.


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Patrice Bertail. Stephan Clémençon. "Bernstein-type exponential inequalities in survey sampling: Conditional Poisson sampling schemes." Bernoulli 25 (4B) 3527 - 3554, November 2019.


Received: 1 July 2017; Revised: 1 September 2018; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110147
MathSciNet: MR4010964
Digital Object Identifier: 10.3150/18-BEJ1101

Keywords: coupling , Esscher transformation , Exponential inequality , Poisson survey scheme , rejective sampling , survey sampling

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4B • November 2019
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