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 theta. Suppose there is also an estimating function m(., mu) identifying another parameter mu via E m(Y,mu) = 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 theta in terms of the hybrid likelihood function H-n(theta) = L-n(theta)(1-a) R-n(mu(theta))(a). Here a is an element of [0,1) represents the extent of the compromise, L-n is the ordinary parametric likelihood for theta, R-n is the empirical likelihood function, and mu 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.