This paper uses the concept of dual likelihood to develop some higher
order asymptotic theory for the empirical likelihood ratio test for
parameters defined implicitly by a set of estimating equations. The
resulting theory is likelihood based in the sense that it relies on
methods developed for ordinary parametric likelihood models to obtain
valid Edgeworth expansions for the maximum dual likelihood estimator
and for the dual/empirical likelihood ratio statistic. In
particular, the theory relies on certain Bartlett-type identities that
can be used to produce a simple proof of the existence of a Bartlett
correction for the dual/empirical likelihood ratio. The paper also
shows that a bootstrap version of the dual/empirical likelihood
ratio achieves the same higher order accuracy as the Bartlett-corrected
dual/empirical likelihood ratio.This
paper is based on Chapter 2 of my Ph.D. dissertation at the University of
Southampton. Partial financial support under E.S.R.C. grant R00429634019 is
gratefully acknowledged. I thank my supervisor, Grant Hillier, for many
stimulating conversations and Peter Phillips, Andrew Chesher, and Jan
Podivisnky for some useful suggestions. In addition, I am very grateful to
the co-editor Donald Andrews and two referees for many valuable comments
that have improved noticeably the original draft. All remaining errors are
my own responsibility.