Estimating the mean of high valued observations in high dimensions

Let Y_i ~ N(μ_i,1), i = 1,…, n, be independent random variables. We study the problem of estimating the quantity S = Σ{i|C<Y_i}μ_i. We emphasize the case where n is large, the vector (μ₁,…,μ_n) is sparse, and the value of C is large. Our approach is nonparametric empirical Bayes, where μ_i are assumed i.i.d from an unknown G. The performance of our suggested estimator is studied both theoretically and through simulations. We also obtain some results related to the local false discovery rates corresponding to high valued points Y_i.