Two-step semiparametric empirical likelihood inference

In both parametric and certain nonparametric statistical models, the empirical likelihood ratio satis es a nonparametric version of Wilks' theorem. For many semiparametric models, however, the commonly used two-step (plug-in) empirical likelihood ratio is not asymptotically distribution-free, that is, its asymptotic distribution contains unknown quantities and hence Wilks' theorem breaks down. This article suggests a general approach to restore Wilks' phenomenon in two-step semiparametric empirical likelihood inferences. The main insight consists in using as the moment function in the estimating equation the in uence function of the plug-in sample moment. The proposed method is general; it leads to a chi-squared limiting distribution with known degrees of freedom; it is e cient; it does not require undersmoothing; and it is less sensitive to the rst-step than alternative methods, which is particularly appealing for high-dimensional settings. Several examples and simulation studies illustrate the general applicability of the procedure and its excellent nite sample performance relative to competing methods.Juan Carlos Escanciano gratefully acknowledges support by the Ministerio Economia y Competitividad (Spain), ECO2017-86675-P & MDM 2014-0431, and by Comunidad de Madrid (Spain), MadEco-CM S2015/HUM-3444. y Ingrid Van Keilegom acknowledges financial support from the European Research Council (2016-2021, Horizon 2020/ERC grant agreement No. 694409)