February 2024 On estimators of the mean of infinite dimensional data in finite populations
Anurag Dey, Probal Chaudhuri
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Bernoulli 30(1): 797-824 (February 2024). DOI: 10.3150/23-BEJ1617

Abstract

The Horvitz-Thompson (HT), the Rao-Hartley-Cochran (RHC) and the generalized regression (GREG) estimators of the finite population mean are considered, when the observations are from an infinite dimensional space. We compare these estimators based on their asymptotic distributions under some commonly used sampling designs and some superpopulations satisfying linear regression models. We show that the GREG estimator is asymptotically at least as efficient as any of the other two estimators under different sampling designs considered in this paper. Further, we show that the use of some well known sampling designs utilizing auxiliary information may have an adverse effect on the performance of the GREG estimator, when the degree of heteroscedasticity present in linear regression models is not very large. On the other hand, the use of those sampling designs improves the performance of this estimator, when the degree of heteroscedasticity present in linear regression models is large. We develop methods for determining the degree of heteroscedasticity, which in turn determines the choice of appropriate sampling design to be used with the GREG estimator. We also investigate the consistency of the covariance operators of the above estimators. We carry out some numerical studies using real and synthetic data, and our theoretical results are supported by the results obtained from those numerical studies.

Acknowledgments

The authors would like to thank Irish Social Science Data Archive (ISSDA) and its administrator, Ruby O’Riordan, for making Electricity Customer Behaviour Trial data available to the authors. The authors gratefully acknowledge careful reading of an earlier version of the paper by two anonymous reviewers. Critical comments and constructive suggestions from the reviewers led to significant improvement of the paper. The authors thank the editor for allowing additional space for presenting several technical details in the paper.

Citation

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Anurag Dey. Probal Chaudhuri. "On estimators of the mean of infinite dimensional data in finite populations." Bernoulli 30 (1) 797 - 824, February 2024. https://doi.org/10.3150/23-BEJ1617

Information

Received: 1 June 2022; Published: February 2024
First available in Project Euclid: 8 November 2023

MathSciNet: MR4665598
zbMATH: 07788904
Digital Object Identifier: 10.3150/23-BEJ1617

Keywords: asymptotic normality , Consistency of estimators , Covariance operator , Heteroscedasticity , high entropy sampling design , inclusion probability , relative efficiency , separable Hilbert space

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Vol.30 • No. 1 • February 2024
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