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 percoordinate 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), 1/n ΣE i=1nlog p(μ i) ≤ η 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. © Institute of Mathematical Statistics, 2006.