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Statistica Sinica 28 (2018), 2389-2407

HYBRID COMBINATIONS OF PARAMETRIC AND
EMPIRICAL LIKELIHOODS
Nils Lid Hjort, Ian W. McKeague and Ingrid Van Keilegom
University of Oslo, Columbia University and KU Leuven

Abstract: 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(, ) identifying another parameter via E𝓂(Y , ) = 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(?) = Ln(?)1-aRn((?))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 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.

Key words and phrases: Agnostic parametric inference, focus parameter, robust methods, semiparametric estimation.

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