Functional central limit theorems for single-stage sampling designs

H. Boistard, H.P. Lopuhaä and A. Ruiz-Gazen,

article published in the Annals of Statistics, vol. 45, n. 4, p. 1728-1758, 2017.

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For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz-Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to generic sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.

Doubly robust inference for the distribution function in the presence of missing survey data

H. Boistard, G. Chauvet and D. Haziza.

Article published in the Scandinavian Journal of Statistics, vol. 43, n. 3, p. 683–699, 2016.

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Item nonresponse in surveys occurs when some, but not all, variables are missing. Unadjusted estimators tend to exhibit some bias, called the nonresponse bias, if the respondents differ from the nonrespondents with respect to the study variables. In this paper, we focus on item nonre- sponse, which is usually treated by some form of single imputation. We examine the properties of doubly robust imputation procedures, which are those that lead to an estimator that remains consistent if either the outcome variable or the nonresponse mechanism are adequately modeled. We establish the double robustness property of the imputed estimator of the finite population distribution function under random hot-deck impu- tation within classes. We also discuss the links between our approach and that of Chambers and Dunstan (1986). The results of a simulation study support our findings.

Approximation of rejective sampling inclusion probabilities and application to high order correlations

H. Boistard, H.P. Lopuhaä and A. Ruiz-Gazen,

Article published in Electronic Journal of Statistics, vol. 6, p. 1967–1983, 2012.
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This paper is devoted to rejective sampling. We provide an expansion of joint inclusion probabilities of any order in terms of the inclusion probabilities of order one, extending previous results by Hájek (1964) and Hájek (1981) and making the remainder term more precise. Following Hájek (1981), the proof is based on Edgeworth expansions. The main result is applied to derive bounds on higher order correlations, which are needed for the consistency and asymptotic normality of several complex estimators.