  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.
