Asymptotic minimaxity of False Discovery Rate thresholding for sparse exponential data

Details Asymptotic minimaxity of False Discovery Rate thresholding for sparse exponential data Asymptotic minimaxity of False Discovery Rate thresholding for sparse exponential data

2006-02-14

arxiv.org/abs/math/0602311v2

Annals of Statistics 2006, Vol. 34, No. 6, 2980-3018

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000920 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000920

We apply FDR thresholding to a non-Gaussian vector whose coordinates X_i, i=1,..., 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 $\log(\mu_i)$, and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of $\log(\mu_i)$: $\frac{1}{n}\sum_{i=1}^n\log^p(\mu_i)\leq \eta^p$. We show for large n and small $\eta$ that FDR thresholding can be nearly Minimax. The FDR control parameter 01/2 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.

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