The Annals of Statistics

Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data

David Donoho and Jiashun Jin

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We apply FDR thresholding to a non-Gaussian vector whose coordinates Xi, i=1, …, n, are independent exponential with individual means μi. The vector μ=(μi) 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(μi), and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of log(μi), $\frac{1}{n}\sum_{i=1}^{n}\,\log^{p}(\mu_{i})\leq \eta^{p}$.

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.

Article information

Ann. Statist. Volume 34, Number 6 (2006), 2980-3018.

First available in Project Euclid: 23 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62C20: Minimax procedures
Secondary: 62G20: Asymptotic properties 62C10: Bayesian problems; characterization of Bayes procedures 62C12: Empirical decision procedures; empirical Bayes procedures

Minimax decision theory minimax Bayes estimation mixtures of exponential model sparsity false discovery rate (FDR) multiple comparisons threshold rules


Donoho, David; Jin, Jiashun. Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data. Ann. Statist. 34 (2006), no. 6, 2980--3018. doi:10.1214/009053606000000920.

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