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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.

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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