Summary: Likelihood and estimating-equation methods provide the two most common approaches to parametric inference. With the former there are three main ways to deal with interval estimation or hypothesis testing; roughly speaking, these are based on the likelihood function, the score function, and the maximum likelihood estimator. For estimating equations which, unlike likelihood, do no require the specification of a full probability distribution for the data, there are analogous methods based on the pseudoscore (estimating function) and on the estimator obtained from the estimating equations.
D. D. Boos [Am. Stat. 46, 327-333 (1992)] has given a lucid review, with emphasis on problems that involve constraints on parameters. The purposes of this paper are to incorporate empirical likelihood into this setting, to compare it with the other methods, and to present some simulations which indicate the need for small-sample corrections in some situations.