×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abramovich, F. and Benjamini, Y. (1995). Thresholding of wavelet coefficients as multiple hypotheses testing procedure. In Wavelets and Statistics. Lecture Notes in Statist. 103 5–14. Springer, New York. · Zbl 0875.62081
[2] Abramovich, F. and Benjamini, Y. (1996). Adaptive thresholding of wavelet coefficients. Comput. Statist. Data Anal. 22 351–361.
[3] Abramovich, F., Benjamini, Y., Donoho, D. and Johnstone, I. (2006). Adapting to unknown sparsity by controlling the false discovery rate. Ann. Statist. 34 584–653. MR2281879 · Zbl 1092.62005
[4] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300. JSTOR: · Zbl 0809.62014
[5] Bretagnolle, J. (1980). Statistique de Kolmogorov–Smirnov pour un enchantillon nonéquiréparti. In Statistical and Physical Aspects of Gaussian Processes (Saint-Flour, 1980) . Colloq. Internat. CNRS 307 39–44. · Zbl 0509.62042
[6] Donoho, D. and Jin, J. (2006). Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data. Technical report, Dept. Statistics, Stanford Univ. Available at arxiv.org/abs/math/0602311. · Zbl 1114.62010
[7] Donoho, D. and Johnstone, I. (1994). Minimax risk over \(\ell_p\)-balls for \(\ell_q\)-error. Probab. Theory Related Fields 99 277–303. · Zbl 0802.62006
[8] Donoho, D. and Johnstone, I. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879–921. · Zbl 0935.62041
[9] Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Statist. 27 642–669. · Zbl 0073.14603
[10] Genovese, C. and Wasserman, L. (2002). Operating characteristics and extensions of the false discovery rate procedure. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 499–517. JSTOR: · Zbl 1090.62072
[11] Jin, J. (2003). Detecting and estimating sparse mixtures. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
[12] Jin, J. (2004). False discovery rate thresholding for sparse data from a location mixture.
[13] Lehmann, E. (1953). The power of rank tests. Ann. Math. Statist. 24 23–43. · Zbl 0050.14702
[14] Lehmann, E. (1986). Testing Statistical Hypotheses , 2nd ed. Wiley, New York. · Zbl 0608.62020
[15] Massart, P. (1990). The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality. Ann. Probab. 18 1269–1283. · Zbl 0713.62021
[16] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[17] Simes, R. (1986). An improved Bonferroni procedure for multiple tests of significances. Biometrika 73 751–754. JSTOR: · Zbl 0613.62067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.