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 $m(\cdot,μ)$ 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 $H_n(θ)=L_n(θ)^{1-a}R_n(μ(θ))^a$. Here $a\in[0,1)$ represents the extent of the compromise, $L_n$ is the ordinary parametric likelihood for $θ$, $R_n$ 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 $a$.

翻译：本文基于参数化似然与非参数化似然的折中，提出了一种混合似然（HL）方法。考虑观测变量$Y$分布的参数化模型，其参数为$θ$。假设同时存在估计函数$m(\cdot,μ)$，通过条件$E\,m(Y,μ)=0$识别另一参数$μ$，该函数在初始定义时独立于参数化模型。为在保持参数化模型优势的同时获得经验似然方法一定程度的稳健性，我们通过混合似然函数$H_n(θ)=L_n(θ)^{1-a}R_n(μ(θ))^a$构建关于$θ$的推断。其中$a\in[0,1)$表示折中程度，$L_n$是$θ$的常规参数似然，$R_n$是经验似然函数，而$μ$通过参数化模型的视角进行考量。我们证明了相应HL估计量的渐近正态性及威尔克斯定理的一种推广形式，并考察了参数化模型误设情况下这些结果的扩展性，最后提出了平衡参数$a$的选择方法。