Maximum likelihood estimation of the log-concave component in a semi-parametric mixture with a standard normal density

The two-component mixture model with known background density, unknown signal density, and unknown mixing proportion has been studied in many contexts. One such context is multiple testing, where the background and signal densities describe the distribution of the $p$-values under the null and alternative hypotheses, respectively. In this paper, we consider the log-concave MLE of the signal density using the estimator of Patra & Sen (2016) for the mixing probability. We show that it is consistent and converges at the global rate $n^{-2/5}$. An EM-algorithm in combination with an active set algorithm implemented in the R-package logcondens was used to compute the log-concave MLE. When one is interested in estimation at a fixed point, a conjecture is made about the limit distribution of our estimator. The performance of our method is assessed through a simulation study.