Hybrid combinations of parametric and empirical likelihoods.

This paper develops a hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods. Consider the setting of a parametric model for the distribution of an observation with parameter . Suppose there is also an estimating function (·, ) identifying another parameter via E(, ) = 0, at the outset defined independently of the parametric model. To borrow strength from the parametric model while obtaining a degree of robustness from the empirical likelihood method, we formulate inference about in terms of the hybrid likelihood function () = () (()) . Here ∈ [0,1) represents the extent of the compromise, is the ordinary parametric likelihood for , is the empirical likelihood function, and is considered through the lens of the parametric model. We establish asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem. We also examine extensions of these results under misspecification of the parametric model, and propose methods for selecting the balance parameter .