A simple and efficient profile likelihood for semiparametric exponential family

Semiparametric exponential family proposed by Ning et al. (2017) is an extension of the parametric exponential family to the case with a nonparametric base measure function. Such a distribution family has potential application in some areas such as high dimensional data analysis. However, the methodology for achieving the semiparametric efficiency has not been proposed in the existing literature. In this paper, we propose a profile likelihood to efficiently estimate both parameter and nonparametric function. Due to the use of the least favorable curve in the procedure of profile likelihood, the semiparametric efficiency is achieved successfully and the estimation bias is reduced significantly. Moreover, by making the most of the structure information of the semiparametric exponential family, the estimator of the least favorable curve has an explicit expression. It ensures that the newly proposed profile likelihood can be implemented and is computationally simple. Simulation studies can illustrate that our proposal is much better than the existing methodology for most cases under study, and is robust to the different model conditions.