Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data

We apply FDR thresholding to a non-Gaussian vector whose coordinates Xi, i=1, …, n, are independent exponential with individual means μi. The vector μ=(μi) 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(μi), and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of log(μi), 1/n∑ᵢ₌₁ⁿ\,logᵖ(μᵢ)≤ ηᵖ. 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.