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 Y with parameter θ. Suppose there is also an estimating function m(·, µ) identifying another parameter µ via E m(Y, µ) = 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 θ in terms of the hybrid likelihood function Hn(θ) = Ln(θ)¹−^aRn(µ(θ))^a. Here a ∈ [0, 1) represents the extent of the compromise, Ln is the ordinary parametric likelihood for θ, Rn is the empirical likelihood function, and µ 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 a.