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An important estimation problem that is closely related to large-scalemultiple testing is that of estimating the null density and the proportion ofnonnull effects. A few estimators have been introduced in the literature;however, several important problems, including the evaluation of the minimaxrate of convergence and the construction of rate-optimal estimators, remainopen. In this paper, we consider optimal estimation of the null density and theproportion of nonnull effects. Both minimax lower and upper bounds are derived.The lower bound is established by a two-point testing argument, where at thecore is the novel construction of two least favorable marginal densities $f_1$and $f_2$. The density $f_1$ is heavy tailed both in the spatial and frequencydomains and $f_2$ is a perturbation of $f_1$ such that the characteristicfunctions associated with $f_1$ and $f_2$ match each other in low frequencies.The minimax upper bound is obtained by constructing estimators which rely onthe empirical characteristic function and Fourier analysis. The estimator isshown to be minimax rate optimal. Compared to existing methods in theliterature, the proposed procedure not only provides more precise estimates ofthe null density and the proportion of the nonnull effects, but also yieldsmore accurate results when used inside some multiple testing procedures whichaim at controlling the False Discovery Rate (FDR). The procedure is easy toimplement and numerical results are given.

Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing