© 1999 by Biometrika Trust
Empirical likelihood in the presence of nuisance parameters
A1 Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213, USA nlazar@stat.cmu.edu A2 Department of Statistics, University of Chicago, Chicago, IL 60637, USA mykland@galton.uchicago.edu
Empirical likelihood was introduced as a nonparametric analogue of ordinary parametric likelihood. It is well known that the empirical likelihood ratio statistic inherits a number of properties of the parametric likelihood ratio statistic, such as the aymptotic chi-squared distribution and Bartlett correctability. This raises the question of whether or not the same is true in the presence of nuisance parameters. Recent work by Qin & Lawless (1994) indicates that the chi-squared distribution is still valid to first order. We show that, when nuisance parameters are present, as introduced via a system of estimating equations, the asymptotic expansion for the signed square root of the empirical likelihood ratio statistic has a nonstandard form. This implies that the empirical likelihood ratio statistic itself does not permit a Bartlett correction.
Keywords:Accuracy; Bartlett correction; Edgeworth expansion; Empirical likelihood; General estimating equation; Likelihood inference; Likelihood ratio test; Martingale inference.