Asymptotic Minimaxity of False Discovery Rate Thresholding for Sparse Exponential Data

Control of the False Discovery Rate (FDR) is an important development in multiple hypothesis testing, allowing the user to limit the fraction of rejected null hypotheses which correspond to false rejections (i.e. false discoveries). The FDR principle also can be used in multiparameter estimation problems to set thresholds for separating signal from noise when the signal is sparse. Success has been proven when the noise is Gaussian; see [3]. In this paper, we consider the application of FDR thresholding to a non-Gaussian setting, in hopes of learning whether the good asymptotic properties of FDR thresholding as an estimation tool hold more broadly than just at the standard Gaussian model. We consider a vector Xi, i = 1, . . . , n, whose coordinates are independent exponential with individual means μi. The vector μ is thought to be sparse, with most coordinates 1 and a small fraction significantly larger than 1. This models a situation where most coordinates are simply ‘noise’, but a small fraction of the coordinates contain ‘signal’. We develop an estimation theory working with log(μi) as the estimand, and use the percoordinate mean-squared error in recovering log(μi) to measure risk. We consider minimax estimation over parameter spaces defined by constraints on the per-coordinate l norm of log(μi): 1 n ( ∑n i=1 log(μi)) ≤ η . Members of such spaces are vectors (μi) which are sparsely heterogeneous. We find that, for large n and small η, FDR thresholding is nearly minimax, increasingly so as η decreases. The FDR control parameter 0 < q < 1 plays an important role: when q ≤ 1 2 , the FDR estimator is nearly minimax, while choosing a fixed q > 1 2 prevents near minimaxity. These conclusions mirror those found in the Gaussian case in [3]. The techniques developed here seem applicable to a wide range of other distributional assumptions, other loss measures, and non-i.i.d. dependency structures.