The Annals of Statistics

Pseudo-Likelihood Theory for Empirical Likelihood

Peter Hall

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It is proved that, except for a location term, empirical likelihood does draw contours which are second-order correct for those of a pseudo-likelihood. However, except in the case of one dimension, this pseudo-likelihood is not that which would commonly be employed when constructing a likelihood-based confidence region. It is shown that empirical likelihood regions may be adjusted for location so as to render them second-order correct. Furthermore, it is proved that location-adjusted empirical likelihood regions are Bartlett-correctable, in the sense that a simple empirical scale correction applied to location-adjusted empirical likelihood reduces coverage error by an order of magnitude. However, the location adjustment alters the form of the Bartlett correction. It is also shown that empirical likelihood regions and bootstrap likelihood regions differ to second order, although both are based on statistics whose centered distributions agree to second order.

Article information

Ann. Statist., Volume 18, Number 1 (1990), 121-140.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing

Bootstrap likelihood confidence interval Cornish-Fisher expansion coverage Edgeworth expansion empirical likelihood pseudo-likelihood second-order correct


Hall, Peter. Pseudo-Likelihood Theory for Empirical Likelihood. Ann. Statist. 18 (1990), no. 1, 121--140. doi:10.1214/aos/1176347495.

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