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Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data. (English) Zbl 1114.62010
Summary: We apply FDR thresholding to a non-Gaussian vector whose coordinates $$X_i$$, $$i=1,\dots,n$$, 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 contains ‘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 $$\log(\mu_i)$$, $$n^{-1}\sum^n_{i=1}\log^p(\mu_i)\leq\eta^p$$.
We show for large $$n$$ and small $$\eta$$ that FDR thresholding can be nearly minimax. The FDR control parameter $$0<q<1$$ plays an important role: when $$q\leq 1/2$$, 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 by F. Abramovich et al. [ibid. 34, No. 2, 584–653 (2006; Zbl 1092.62005)]. The techniques developed here seem applicable to a wide range of other distributional assumptions, other loss measures and non-i.i.d. dependency structures.

MSC:
 62C20 Minimax procedures in statistical decision theory 62G20 Asymptotic properties of nonparametric inference 62C10 Bayesian problems; characterization of Bayes procedures 62H12 Estimation in multivariate analysis 62C12 Empirical decision procedures; empirical Bayes procedures 62J15 Paired and multiple comparisons; multiple testing
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References:
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