Empirical Likelihood Ratio Confidence Intervals for Unequal Probability Sampling
Yves G. Berger1, Omar De La Riva2
1Southampton Statistical Sciences Research Institute, University of Southampton, Southampton, England, United Kingdom; 2Social Statistics, University of Southampton, Southampton, England, United Kingdom

There have been many recent developments of empirical likelihood based methods (Hartley & Rao 1962, Owen, 1988) in survey sampling (e.g. Rao & Wu 2009) since Chen and Qin (1993) suggested its first application in survey sampling. Chen and Sitter (1999) proposed a pseudo empirical likelihood approach which can be used to construct confidence intervals for the Hájek (1971) ratio estimator (Wu & Rao, 2006). However, from a theoretical point of view, the pseudo empirical likelihood approach is not entirely satisfactory, as it is not a genuine empirical likelihood approach, and it is not applicable to the Horvitz-Thompson (1952) estimator. We propose an empirical likelihood approach for unequal probability sampling without replacement which can be implemented to construct confidence intervals for the Horvitz-Thompson estimator. We show that the profile empirical likelihood function has asymptotically a chi-square distribution under a set of regularity conditions. We support our results with a simulation study.


Chen, J. & Qin, J. (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika 80, 107-116.

Chen, J. & Sitter, R.R. (1999). A pseudo empirical likelihood approach to the effective use of auxiliary information in complex surveys. Statistica Sinica 9, 385-406.

Hájek, J. (1971). Foundations of Statistical Inference. Toronto, Canada: Holt, Rinehart, Winston. Chap. Discussion of an essay on the logical foundations of survey sampling, part on by D. Basu.

Hartley, H.O. & Rao, J.N.K. (1962). Sampling with unequal probabilities and without replacement. Annals of Mathematical Statistics 33, 350-374.

Horvitz, D.G. & Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association 47, 663-685.

Owen, A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237-249.

Rao, J.N.K. & Wu, C. (2009). Empirical Likelihood Methods. In: D. Pfeffermann and C.R. Rao. (editors). Elsevier.

Wu, C. & Rao, J.N.K. (2006). Pseudo-empirical likelihood ratio confidence intervals for complex surveys. The Canadian Journal of Statistics 34, 359-375.

Résumé: Nous proposons une approche du type vraisemblance empirique pour estimer l'intervalle de confiance de l'estimateur de Horvitz-Thompson (1952) pour un plan de sondage à probabilité inégales sans remise. Nous montrons que sous des conditions de régularité, le profile de la vraisemblance empirique a une distribution chi carré. Nous présenterons également quelques résultats de simulation.

Keywords: Horvitz-Thompson estimator; Inclusion probabilities; Profile Likelihood; Sample Surveys

Biography: Omar De La Riva has a Bachelor degree in Actuarial Sciences and a Master degree in Probability and Statistics from the National Autonomous University of Mexico. Omar research interest focuses on Statistics and Survey Sampling methodology. He is currently a PhD student at the University of Southampton. The topic of his PhD thesis is on application of empirical likelihood in survey sampling estimation. Omar has worked for Mexican government agency and educational entities as a Statistical consultant and Head of Statistics Department.