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 $m(\cdot ,\mu )$ identifying another parameter $\mu$ via $Em(Y,\mu )=0$, 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 $a\in [0,1)$ 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 $a$.