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 Em(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-a Rn (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.

中文翻译：

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参数和经验似然的混合组合

本文基于参数和非参数似然之间的折衷开发了一种混合似然 (HL) 方法。考虑为具有参数 θ 的观测值 Y 的分布设置参数模型。假设还有一个估计函数 m(·, μ) 通过 Em(Y, μ) = 0 识别另一个参数 μ，在开始时独立于参数模型定义。为了从参数模型中借用强度，同时从经验似然方法中获得一定程度的稳健性，我们根据混合似然函数 Hn (θ) = Ln (θ)1-a Rn (μ(θ)) 来制定关于 θ 的推理一种 。这里 a ∈ [0,1) 表示折衷的程度，Ln 是 θ 的普通参数似然，Rn 是经验似然函数，μ 是通过参数模型的镜头来考虑的。我们建立了相应 HL 估计量的渐近正态性和 Wilks 定理的一个版本。我们还在参数模型的错误指定下检查了这些结果的扩展，并提出了选择平衡参数 a 的方法。