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An empirical likelihood approach under cluster sampling with missing observations

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  • Yves G. Berger

    (University of Southampton)

Abstract

The parameter of interest considered is the unique solution to a set of estimating equations, such as regression parameters of generalised linear models. We consider a design-based approach; that is, the sampling distribution is specified by stratification, cluster (multi-stage) sampling, unequal selection probabilities, side information and a response mechanism. The proposed empirical likelihood approach takes into account of these features. Empirical likelihood has been mostly developed under more restrictive settings, such as independent and identically distributed assumption, which is violated under a design-based framework. A proper empirical likelihood approach which deals with cluster sampling, missing data and multidimensional parameters is absent in the literature. This paper shows that a cluster-level empirical log-likelihood ratio statistic is pivotal. The main contribution of the paper is to provide the rigorous asymptotic theory and underlining regularity conditions which imply $${\surd {n}}$$√n-consistency and the Wilks’s theorem or self-normalisation property. Negligible and large sampling fractions are considered.

Suggested Citation

  • Yves G. Berger, 2020. "An empirical likelihood approach under cluster sampling with missing observations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(1), pages 91-121, February.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:1:d:10.1007_s10463-018-0681-x
    DOI: 10.1007/s10463-018-0681-x
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    References listed on IDEAS

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    1. Yves G. Berger & J. N. K. Rao, 2006. "Adjusted jackknife for imputation under unequal probability sampling without replacement," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 531-547, June.
    2. Song Chen & Ingrid Van Keilegom, 2009. "A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 415-447, November.
    3. Qihua Wang & J. N. K. Rao, 2002. "Empirical Likelihood‐based Inference in Linear Models with Missing Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 563-576, September.
    4. D. Pfeffermann & C. J. Skinner & D. J. Holmes & H. Goldstein & J. Rasbash, 1998. "Weighting for unequal selection probabilities in multilevel models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 23-40.
    5. Guido W. Imbens & Tony Lancaster, 1994. "Combining Micro and Macro Data in Microeconometric Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 61(4), pages 655-680.
    6. Qihua Wang, 2002. "Empirical likelihood-based inference in linear errors-in-covariables models with validation data," Biometrika, Biometrika Trust, vol. 89(2), pages 345-358, June.
    7. Wayne A. Fuller, 2009. "Some design properties of a rejective sampling procedure," Biometrika, Biometrika Trust, vol. 96(4), pages 933-944.
    8. David Haziza & Jean‐François Beaumont, 2007. "On the Construction of Imputation Classes in Surveys," International Statistical Review, International Statistical Institute, vol. 75(1), pages 25-43, April.
    9. Qin, Jing & Zhang, Biao & Leung, Denis H. Y., 2009. "Empirical Likelihood in Missing Data Problems," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1492-1503.
    10. Song Chen & Ingrid Van Keilegom, 2009. "Rejoinder on: A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 468-474, November.
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    Cited by:

    1. Berger, Yves G. & Patilea, Valentin, 2022. "A semi-parametric empirical likelihood approach for conditional estimating equations under endogenous selection," Econometrics and Statistics, Elsevier, vol. 24(C), pages 151-163.

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