Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data

Abstract

We apply FDR thresholding to a non-Gaussian vector whose coordinates X_i, i=1, …, n, are independent exponential with individual means μ_i. The vector μ=(μ_i) is thought to be sparse, with most coordinates 1 but a small fraction significantly larger than 1; roughly, most coordinates are simply ‘noise,’ but a small fraction contain ‘signal.’ We measure risk by per-coordinate mean-squared error in recovering log(μ_i), and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of log(μ_i), $\frac{1}{n}\sum_{i=1}^{n}\,\log^{p}(\mu_{i})\leq \eta^{p}$.

We show for large n and small η that FDR thresholding can be nearly minimax. 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 Abramovich et al. [Ann. Statist. 34 (2006) 584–653]. The techniques developed here seem applicable to a wide range of other distributional assumptions, other loss measures and non-i.i.d. dependency structures.