Asymptotic Minimaxity of False Discovery Rate Thresholding for Sparse Exponential Data

We apply FDR thresholding to a non-Gaussian vector whose coordinates $X_{i},i=1,\ldots ,n$, are independent exponential with individual means $\mu _{i}$. The vector $\mu =(\mu _{i})$ 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 ${\rm log}(\mu _{i})$, and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of ${\rm log}(\mu _{i})$, ${\textstyle\frac{1}{n}}\sum_{i=1}^{n}{\rm log}^{p}(\mu _{i})\leq \eta ^{p}$. We show for large n and small . 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.