A Bernstein-type estimator for decreasing density with application to pp-value adjustments

The nonparametric maximum likelihood estimator (NPMLE) is a popular approach to estimating decreasing densities, i.e., f(s)≥f(t),s≤tf(s)≥f(t),s≤t. A less ideal feature of NPMLE is its step-function form. In this paper, we propose two nonparametric density estimators based on the Bernstein-type polynomials of the NPMLE. The proposed estimators have relatively simple forms and easy implementation. They have satisfactory smoothness as well as estimation efficiency. Numerical examples demonstrate the superior performance of the proposed estimators compared to existing methods. Decreasing densities have been applied in simultaneous inference to estimate the proportion of true null hypotheses and the local false discovery rate. We applied the proposed estimators to conduct simultaneous tests for a gene expression data set.