Optimal Rates of Convergence for Estimating the Null Density and Proportion of Nonnull Effects in Large-Scale Multiple Testing

Thumbnail Image
Penn collection
Statistics Papers
Degree type
characteristic function
empirical characteristic function
Fourier analysis
minimax lower bound
multiple testing
null distribution
proportion of nonnull effects
rate of convergence
two-point argument
Statistics and Probability
Grant number
Copyright date
Related resources
Cai, T. Tony
Jin, Jiashun

An important estimation problem that is closely related to large-scale multiple testing is that of estimating the null density and the proportion of nonnull effects. A few estimators have been introduced in the literature; however, several important problems, including the evaluation of the minimax rate of convergence and the construction of rate-optimal estimators, remain open. In this paper, we consider optimal estimation of the null density and the proportion of nonnull effects. Both minimax lower and upper bounds are derived. The lower bound is established by a two-point testing argument, where at the core is the novel construction of two least favorable marginal densities f1 and f2. The density f1 is heavy tailed both in the spatial and frequency domains and f2 is a perturbation of f1 such that the characteristic functions associated with f1 and f2 match each other in low frequencies. The minimax upper bound is obtained by constructing estimators which rely on the empirical characteristic function and Fourier analysis. The estimator is shown to be minimax rate optimal. Compared to existing methods in the literature, the proposed procedure not only provides more precise estimates of the null density and the proportion of the nonnull effects, but also yields more accurate results when used inside some multiple testing procedures which aim at controlling the False Discovery Rate (FDR). The procedure is easy to implement and numerical results are given.

Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
Journal title
The Annals of Statistics
Volume number
Issue number
Publisher DOI
Journal Issue
Recommended citation