An Unified Theory of Empirical Likelihood Confidence Intervals under Unequal Probability Sampling from a Finite Population

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Description/Abstract

A proper empirical likelihood approach for confidence intervals under unequal probability sampling is non-existent in the literature. We propose a proper empirical likelihood approach which can be used to construct design-based confidence intervals under unequal probability sampling. The proposed approach gives confidence intervals which may have better coverages than standard confidence intervals and pseudo empirical likelihood confidence intervals, which rely on variance estimates and design-effects. The proposed approach does not rely on variance estimates, design-effects, re-sampling or linearization, even when the parameter of interest is not linear. It can be also used to construct confidence intervals of means, regressions coefficients, quantiles, totals or counts even when the population size is unknown. It also gives suitable confidence intervals when the point estimator is biased. We show that the proposed maximum empirical likelihood estimator is asymptotically optimal. We also propose an approach which deals with large sampling fractions. It also offers a likelihood-based justification for design-based approaches, such as calibration, used in sample surveys. We compare the proposed approach with the pseudo empirical likelihood approach, which is not a proper empirical likelihood approach, because its empirical log-likelihood ratio function does not converge to a chi-square distribution. For confidence interval, the pseudo empirical log-likelihood ratio function needs to be adjusted by a factor (the design effect) which need to be estimated. This may affect the coverages of the confidence intervals. We also apply the proposed approach to a measure of poverty based upon the European Union Survey on Income and Living Conditions (EU-SILC).