  We extend the empirical likelihood of Owen [Ann. Statist. 18 (1990) 90-120]
by partitioning its domain into the collection of its contours and mapping the
contours through a continuous sequence of similarity transformations onto the
full parameter space. The resulting extended empirical likelihood is a natural
generalization of the original empirical likelihood to the full parameter
space; it has the same asymptotic properties and identically shaped contours as
the original empirical likelihood. It can also attain the second order accuracy
of the Bartlett corrected empirical likelihood of DiCiccio, Hall and Romano
[Ann. Statist. 19 (1991) 1053-1061]. A simple first order extended empirical
likelihood is found to be substantially more accurate than the original
empirical likelihood. It is also more accurate than available second order
empirical likelihood methods in most small sample situations and competitive in
accuracy in large sample situations. Importantly, in many one-dimensional
applications this first order extended empirical likelihood is accurate for
sample sizes as small as ten, making it a practical and reliable choice for
small sample empirical likelihood inference.
